merton lifetime portfolio selection 1969

7/27/2019 Merton Lifetime Portfolio Selection 1969

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LIFETIME PORTFOLIO SELECTION UNDERUNCERTAINTY: THE CONTINUOUS-TIME CASE

Rober t C.I IntroductionM O ST m odels of po rtfolio select ion havebeen one-period models. I examine thecombined problem of optimal portfolio selec-t ion a nd consum ption rules for an individual in

a continuous-t ime model where his income isgenerated by returns on assets and these re-turns or instantaneous "growth rates" are sto-chast ic. P . A. Samuelson has developed a sim-ilar model in discrete-time for more generalprobabil i ty distr ibutions in a companion paperC81.I derive the optimali ty eq uations for a m ult i-asset problem when the rate of returns aregenerated by a Wiener Brownian-motion proc-ess. A par ticul ar case examined in detail isthe two-asset model with con stant relat ive r isk-aversio n or iso-elastic ma rginal utility. Anexplicit solution is also found for the case ofconsta nt absolute risk-aversion. T he generaltechnique employed can be used to examine awide class of intertemporal economic problemsunder uncertainty.I n addi t ion to the Samuelson paper [S] , thereis the multi-period analysis of Tobin [9].Phelps [6 ] h as a model used to determine theoptimal consumption rule for a multi-periodexample where income is part ly generated byan asset with an uncer tain return . Mirr less [ S ]has developed a continuous-t ime optimal con-sum ption model of the neoclassical type withtechnical progress a random variable.I1 D ynamics of the Model: Th e Budg et Equa tion

In the usual continuous-t ime model undercertainty, the budget equation is a differentialequa tion. How ever, when unce rtainty is intro-duced by a random var iable, the budget equa-* T h i s w o r k was done during the tenure of a Nat ionalDeiense Educat ioi l Act Fel lowship. ;\id f ro m t he Na t iona lScience Foundat ion is gra teful ly acknowledged. I a m i n -d e b t e d t o P a u l -4. amuelson for many discuss ions andhis helpful suggestions. I wish to thank Stanley Fischer ,Massachusetts I ns t i tu te of Technology, for his comments

on sec t ion 7 an d John S. Flemm ing for his cr it ic ism of anearlier version.

t ion m ust be general ized to become a stochasticdifferential equation. T o see the meaning ofsuch an equation, i t is easiest to work out thediscrete-t ime version and then pass to the l imitof continuous time.DefineW ( t ) =X i ( t ) =C ( t ) =w & ( t )=

total wealth at time tprice of the it hasset at time t , ( i = 1 ,. . , )consumption per unit time at time tproportion of total wealth in the i&asset at time t , ( i = 1 , . . ,m )So te

i=

The budget equat ion can be wri t ten as

where t = o+ h an d the t ime in terval betweenperiods is h. B y subt ract ing W ( t , ) f rom bothnzsides a nd using ): w , ( t o ) = 1 , we can rewrite

i = 1(1) as ,

the rate of return per unit t ime on the ith sset .T h e g , ( t ) are assumed to be generated by astochastic process.I n d iscrete t ime, I make the fur ther assump-t ion that g , ( t ) is determined as fol lows,g L ( t ) h ( a , - u L 2 / 2 ) h AY, ( 3 )

where ai,he "expected" rate of return, is con-


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stant; and Yi(t) is generated by a Gaussianrandom-walk as expressed by the stochasticdifference equation,~ , ( t ) y,(to)= n y , = u , z , ( ~ )~ (4 )

where each Z,(t) is an independent variatewith a standard normal distribution for everyt,ui2 s the variance per unit time of the processY, and the mean of the increment .A Y, is zero.Substituting for g,(t ) from ( 3 ) , we can re-write (2) as,mW(t) -W(tO)= 2 wi(to) e(az-c2/2(h+AY)- 1)1 (W(t0) - C(t0)h)- C(to)h. (5)

Before passing in the limit to continuoustime, there are two implications of (5) whichwill be useful later in the paper.

and

(AY,AY,) .W2(to)+O(h2)( 7 )where E(to) is the conditional expectationoperator (conditional on the knowledge ofW(to)), and 0 a ) is the usual asymptotic ordersymbol meaning "the same order as."The limit of the process described in (4) ash -+ 0 (continuous time) can be expressed bythe formalism of the stochastic differentialequation,l

dYd = u,Z,(t) ddt (4')and Y, ( t ) is said to be generated by a Wienerprocess.By applying the same limit process to thediscrete-time budget equation, we write (5) as

The stochastic differential equation (5') is thegeneralization of the continuous-time budgetequation under uncertainty.'See K. I t 8 [41, for a rigorous discussion of stochastic

A more familiar equation would be the aver-aged budget equation derived as follows: From(S), we have

- C(to)h] - C(to)+O(h). ( 8 )Now, take the limit as h -+ 0, so that (8) be-comes the following expression for the defined"mean rate of change of wealth":0W (to)= imit E (to)def. h+ O 1

I11 The Two-AssetModelFor simplicity, I first derive the optimalequations and properties for the two-assetmodel and then, in section 8, display the gen-eral equations and results for the m-asset case.Define

2 48 THE REVIEW OF ECONOMICS AND STATISTICS

differential equations. as follows,

wl (t)=w (t ) = proportion invested in therisky assetwz(t) = 1-w(t) = proportion invested in thesure assetgl (t ) = d t ) = return on the risky assetW ar gl > 0)g2(t) = r = return on the sure asset(Var g2 = 0)Then, for g(t)h = (a-u2/2) h + AY, equa-tions (51, ( 6 ) , 7 ) , and (8') can be written as,

W(t>- W(t0)- [w(t ,) (e(a-@/2(h+AY)- 1)


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LIFETIME PORTFOLIO SELECTION 249E {$be-pt ULC(t)]dt + B [ W ( T ) , T ]

( 1 2 ) , (14)2 0 ; W ( t )> 0 ; W ( 0 )= Wo > 0

U ( C ) is assumed to be a strictlyU'(C) > 0 ;< 0 ) where g ( t ) is a random variable

B [ W ( T ) , T ] s to be a specified "be-(also referred to in" and usually assumed to be concave in

"E" in (14) is short for E( O) , theW (0)o as known.

In ( 17) , take the E ( t o ) operator onto eachterm and, noting that I IW(t , ) , to] = E ( t o )I [ W ( t o )to] , subtract I I W ( t o ) o] from bothsides. Substitute from equations ( 10) and ( 1 1 )for E(t0)[ W ( t )- W( t 0) l and E(t0) l(W(t)- W ( t o ) ) 2 ] ,nd then divide the equation byh. Take the limit of the resultant equation ash* and ( 17 ) becomes a continuous-time ver-sion of the Bellman-Dreyfus fundamental equa-tion of optimality, ( 17').

0 = Max art[ e-pt u [ C ( t ) +-C(t>,w(t ) at

o derive the optimality equations, I restate a2rt t ) W 2 ( t ) ]) in a dynamic programming form so that + FUw ( ( 17')principle of optimality can bed. To do this, define, where It is short for I [ W ( t ) , t ]and the sub-r script on to has been dropped to reflect that=Max E ( t ) e-ps U[C(s)lds (17') holds for any t E r0,T1.

{C(s) ,w(s) I+ B [ W ( T ) , T l ] ( 1 5 )( 1 5 ) is subject to the same constraints(14) . Therefore,

= B [ W ( T ) , T l . ( 1 5 ' )n general, from definition ( I S ) ,( t o )to]= Max E ( t o ) e-ps U[C s ) ds{C(s ) ,w(s )>+ I [W( t ) l t l] ( 1 6 )

(14) can be rewritten asI(Wo,O)= MaxE[$t, e-pS U [C (s )] ds{C(s) ,w(s>>+ I W ( t ) , t l . (14')

t = o+ h and the third partial derivativesI IW ( to) , to] re bounded, then by Taylor's

( 1 6 ) can be rewritten as

- . -If we define +(w,C;W ; t ) = { e-pt U ( C )art+ - a ] ,at +- ( w ( t ) ( a- r ) + r ) W ( t )-aw

(17') can be written in the more compact form,Max + (w,C;W, t ) = 0.{C,w>

The first-order conditions for a regular interiormaximum to (17") are,+ C [w*,C*;W,:t]= 0 = e-pt U'(C) - aZt/aW

and ( 1 8 )art+w [w*;C*;W;t] 0 = (a-r)-wIIW(to,to]= Max E(to) { e - p i u [ ~ ( t ) ] A set of sufficient conditions for a regular in-{ C Y> a W V o > , t o l terior maximum is+ LW(to),tol + at 9"" < 0 ; +c c < 0 ; det ['"" S w C ] > 0.

a r [ w ( t o ) , t o l [ W ( t )- W ( t 0 ] + o w +C C+ aw (btcc= (bow = 0, and if I [ W ( t ) , t ]were strictly

1 a 2 ~ [ ~ ( t o ) , t o ] concave in W , hen+T 3w2 $00 = Ur'(c)< 0, by the strict concavity of U-

[ w ( t ) w ( t ~ ) 1 2 + O ( l r Z ) ) and (20)f E [ t o $ ] . ( 1 7 )

The basic derivation of the optimality equations in this a @ ( w , C : : ) is short for the rigorous @ [ w , C ; I , / a t ;S. E. Dreyfus [21, Chapter VII. artlaw;a z l t / m ; ~ : ; t l .


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250 T H E R E V IE W O F E C ON O M IC S A N D S T AT IS T IC Sa21t(Pww = W(t)o2- 0 4 , by the strict con-aw2cavity of It, (21)

and the sufficient conditions would be satisfied.Thus a candidate for an optimal solution whichcauses Z[W (t), t] to be strictly concave will beany solution of the conditions (1 7')-(2 1).The optimality conditions can be re-writ tenas a set of two algebraic an d one partia l dif-ferential equation to be solved for w* (t) , C *( t) ,a nd I [ W ( t ) , t ] .

+[w*,C*;W;t] = 0 (17")+c[w*;C*;W;t] = 0+,[w*,C*;W;t] = 0 (19)subject to the boundary conditionI [ W ( T ) , T ] = B [W (T) , T] an dthe solution being a feasible solu-tion to (14).IV Constant Relative Risk A version

The sys tem (*) of a nonlinear partial dif-ferential equation coupled with two algebraicequations is difficult to solve in general. How-ever, if t he u tility function is assumed to be ofthe form yielding constant relative risk-aver-sion (i,e., iso-elastic ma rginal ut ility ), then (*)can be solved explicitly. Therefore, let U(C)= Cy/y, y < 1 an d y f . 0 o r U(C ) = log C(the limiting form for y = 0) where - U"(C)C/U'(C) = 1 - y = i s Prat t ' s [7] measureof relative risk aversion. T he n, system (*) canbe writ ten in this particular case as

I ubject to I [W (T ) TI = 2 - y e-pT[ W ( T ) ] Y / ~ ,or 0 < E < < 1 J4 B y the substitution of the results of (18 ) into (1 9) a t(C*,w*), we have the condi t ion w*(t)(a - ) > 0 if andThe paper considers only interior optimal solutions. Theproblem could have been formulated in the more general

Kuh n-Tuck er fo rm in which case the equalities of (18 ) and(19) would be replaced with inequalities.

where a strategically-simplifying assumptionhas been made as to the particular form of thebequest valuation function, B [ W ( T ) T I 6T o solve (17") of (*'), take as a trial solu-tion,

By substitution of th e trial solution into ( 17"),a necessary condi t ion that I , [W(t) , t ] be asolution to (17") is found to be that b (t ) m ustsatisfy th e following ordina ry differential equa-tion,d ( t ) = ,ab(t) - ( l - y ) [ b ( t ) ] - y l l - v (23)sub jec t to b (T) = 2 - 7 , and where p = p - y

[(a- ( 1 - y ) + r] . T he resulting de-cision rules for consumption and portfolio selec-t ion , C*(t ) and w*(t) , are from equat ions(18) and (19) of (*'), thenC*( t ) = [b(t )I1 ly-I W (t) (24)and

T he solution to (23 ) isb ( t ) = { [ l + (VC -1 )eY(t-T)]/v}l-v (26)where v = / ( 1 - y ) .A sufficient condition for I [ W ( t ) , t ] to be asolution to (*') is tha t Z[W (t ) , t ] s a ti s fyA. f [W (t ), t] be real (feasibility)a2ft


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LI FETI ME PORTFOLIO SELECTION 251C+(t)= [v/(l + (ve-I) e"t-T))] W( t) ,for v fi= [ l / ( T - t + e ) ] W ( t ) , f o r v = O(28)ndw*(t) = (a-y) =w*, a constant independ-02(1-y)ent of W or t. (29)

V Dynamic Behavior and the BequestValuation FunctionThe purpose behind the choice of the par-(*') wasily mathematical. The economic motivethat the "true" function for no bequests is

= 0 (i.e., E = 0). From (28),= T when l = 0.

to examine the dynamic behavior of C*( t)"error7' or an implicit par t of the eco-requirements of the problem, the param-

l was introduced.re 1, (C*/W) t = ~ w as E+ 0. How-

To examine some of the dynamic propertiesof C*(t), let E = 0, and define V(t) = C*(t)/W (t) 1 , the instantaneous marginal (in thiscase, also average) propensity to consume outof wealth. Then, from (28),k( t ) = [V(t)]"v(t-T) (30)

and, as observed in figure 1 (for E = 0), V(t)is an increasing function of time. In a generali-zation of the half-life calculation of radioactivedecay, define T as that te [O,T] such tha t V(T)= nV(0) (i.e., T is the length of time requiredfor V(t) to grow to n times its initial size).Then, from (28),

, for v = 0.(3 1)To examine the dynamic behavior of W(t)under the optimal decision rules, it only makessense to discuss the expected or "averaged"behavior because W(t) is a function of a ran-dom variable. To do this, we consider equation(13), the averaged budget equation, and evalu-ate it a t the optimal (w*,C*) to form

rate of consumption. Because there is zeroutility associated with positive wealth for t >T, the mathematics reflects this by requiringthe optimal solution to drive W(t) + 0 ast+T. Because C* is a flow and W(t) is astock and, from (28), C* is proportional toW( t), (C*/W) must become larger and largeras t+T to make W(T) = 0.7 In fact, ifW(T) - ) > 0, an "impulse" of consumptionwould be required to make W(T) = 0. Thus,equation (28) is valid for l = 0.'The problem described is essentially one of exponentialdecay. If W( t )= f ( t )> 0, finite for all t , andWo> 0, then it will take an infinite length of time forW( t )= 0. However, if f ( t )+ eel as t+ T, then W( t )+as t+ T.

, ,(a- l2where a, = + r ] , and, in sec-

tion VII, a, will be shown to be the expectedreturn on the optimal portfolio.Ey differentiating (13') and using (30) , weget

which implies that for all finite-horizon optimalpaths, the expected rate of growth of wealth isa diminishing function of time. Therefore, ifa, < V(O), the individual will dis-invest (i.e.,he wiIl plan to consume more than his expectedincome, a,W(t) ). If a. > V(O), he will planto increase his wealth for 0 < t < f , and then,dis-invest at an expected rate a, < V(t) fori < t < T where f is defined as the solution to

Further, ai/aa, > 0 which implies that thelength of time for which the individual is a net


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252 THE REVIEW OF ECONOMICS AND STATISTICSsaver increases with increasing expected re-turns on the portfolio. Thus, in the case a. >V(O), we find the familiar result of "humpsaving."

VI Infinite Tim e HorizonAlthough the infinite time horizon case ( T =w) yields essentially the same substantive re-sults as in the finite time horizon case, it isworth examining separately because the opti-mality equations are easier to solve than forfinite time. Therefore, for solving more com-plicated problems of this type, the infinite timehorizon problem should be examined first.The equation of optimality is, from section111,

However (17') can be greatly simplified byeliminating its explicit time-dependence. De-fineJ[W(t),t]= ptIIW(t),t]= Max E(t) S,"e-p(~-~)[C]ds{CP>= Max B S; e-pv U[C] dv,

{C,w>independent of explicit time. (34)Thus, write J[W(t),t] = J[W] to reflect thisindependence. Substituting J [W] , dividing bye-pt, and dropping all t subscripts, we can re-write (17') as,0 = Max [U(C) - PJ+ J'(W).{CP>

{(w(t) a-r) + r)W-C)+ 1/2 J"( W) a2w2W2]. (35)Note: when (35) is evaluated at the optimum(C*,w*) , it becomes an ordinary differentialequation instead of the usual partial differentialequation of (17'). For the iso-elastic case,( 3 5 ) can be written as"Hump saving" has been widely discussed in the litera-ture. (See J. D e V. Graaff [31 for such a discussion.)Usually "hump saving" is discussed in the context of workand retirement periods. Clearly, such a phenomenon canoccur without these assumptions as the example in this

paper shows.

where the functional equations for C* and w*have been substituted in equation (36).The first-order conditions corresponding to(18) and (19) are0 = U'(C) - '(W) ( 3 7 )and0 = (a-r)J'(W) + JUwWn2 (38)and assuming that limit B [W(T) TI = 0, theT+mboundary condition becomes the transversalitycondition,limit E[I[W (t) ,t ] = 0t+ m

(39)orlimit E[e-pt J[ W(t) ] ] = 0t+-which is a condition for convergence of the in-tegral in (14). A solution to (14) must satisfy(39) plus conditions A, B, and C of section IV.Conditions A, B, and C will be satisfied in theiso-elastic case if

> 0 (40)holds where (40) is the limit of condition ( 2 7)in section IV, as T+ and V* = C*(t)/W(t)when T = w. Condition (39) will hold if p >y $/w where, as defined in (13), +( t) is thestochastic time derivative of W (t) and ~ ( t ) /W(t) is the "expected" net growth of wealthafte r allowing for consumption. That (39) issatisfied can be rewritten as a condition on thesubjective rate of time preference, p, as follows:for y < 0 (bounded utility), P > O

y = 0 (Bernoulli log case), P > o0 < y < 1 (unbounded utility), P > v

(41)Condition (41) is a generalization of the usualassumption required in deterministic optimalconsumption growth models when the produc-tion function is linear: namely, that p > Max[0, y P] where P = yield on capita1.O If a "di-'If one takes the limit as 0 (where T,' is the

variance of the composite portfolio) of condition (41), hen


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L I F E T I M E P O R T F O L I O S E L E C T I O N 2 53minishing-returns," strict ly-concave '(produc-tion" function for wealth were introduced, thena positive p would suffice.If condition (41) is satisfied, then condition(4 0) is satisfied. Th ere for e, if i t is assum edtha t p satisfies (41), then th e rest of the deri-vation is the same as for the finite horizon casean d the optimal decision rules are,

and

T h e ordinary differential equation (3 5 ),J" = f( J, J ' ) , has "extraneous" solutions otherthan the one tha t genera tes (42) and (43) .However, these solutions are ruled out by thetransversal i ty condit ion, (39), and condit ionsA, B, and C of section IV . As was expected,l imi t C*(t ) = C,*(t) an d l imi t w* (t ) = wm*T + m T + m( t ) .T h e main purpose of th is section was toshow that the partial differential equation (17')can be reduced in the case of infinite timehorizon to a n o rdinary differential equation.

VII Economic Interpretation of the OptimalDecision Rules for Portfolio Selectionand ConsumptionAn important result is the confirmation ofthe theorem proved by Samuelson [8] , for thediscrete-time case, stating that, for iso-elasticmarginal utility, the portfolio-selection deci-sion is independe nt of the cons ump tion deci-sion. Fu rth er, for the special case of Bernoulli

logarithm ic utility (y = O), the separa t ion goesboth ways, i .e., the consumption decision isindepen dent of th e financial para me ters an d isonly dependent upon the level of wealth. Thisis a result of two assumptions: (1 ) constantrelative risk-aversion (iso-elastic marginal util-i ty) which implies that one's at t i tude towardfinancial risk is independent of one's wealthlevel, and ( 2 ) the stochastic process which(41) becomes the condit ion that p >max[O,ya*l where a*is the yield on the composite por tfol io. Thu s , the deter -ministic case is the limiting form of (41 ).

generates the price changes (independent in-crements assumption of the Wiener process).With these two assumptions, the only feed-backs of the system, the price change an d theresulting level of wealth, have zero relevancefor the portfolio decision and hence, it is con-stant .The optimal proport ion in the risky asset ,1w*, can be rewritten in terms of P rat t 's relativerisk-aversion measure, 6, as

The qua l i t a t ive re su l t s t ha t aw*/aa > 0,a w * / a r < 0 , a w * / a ~ V , a n d a w * / a 8 < 0are intuit ively clear and need no discussion.However, because the optimal portfolio selec-tion rule is constant, one can define the opti-mum composite portfol io and i t wil l have aconstant mean and var iance . Namely,

= Var [ w * ( a + A Y ) + (1 -w*)Y]

After having determined the optimal w*, onecan now think of t he original problem a s havingbeen reduced to a simple Phelps-Ramsey prob-lem, in which we seek an optimal consumptionrule given that income is generated by theuncertain yield of an (composite) asset.Thus, the problem becomes a continuous-t ime analog of the one examined by Ph elps [6 ]in discrete time. Therefore, for consistency,C,* (t) should be expressible in term s of a* ,u.', 6, p , and W ( t) only. T o show th at th is is ,in fact , the result , (42) can be rewrit ten as, ' '

10 Note : no res tric t ion o n borrowing or going sho r t wasimposed on the problem, and therefore, w* can be greatertha n one or less than zero. Thu s , if a < , the r isk-aver termill sh or t some of th e r isky asset, an d if a > f 2 8 , hewil l bor row funds to invest in the r isky asset . If one wishedto res tr ic t w*e[O, lI , then such a constraint could be intro-duced and handled by the usua l K uhn- Tucker me thodswith resulting inequalities.Because this section is concerned with the qualitativechanges in the solut ion with respect to shif ts in the param-eters, the more-simple form of the infinite-time horizoncase is examined. T he essential difference betw een C ,*(t)and C*( t) is the explici t t ime dependence of C * ( t ) whichwas discussed in section 1.. For s implici ty, the "m " onsubsc ript C,*(t) will be deleted for the rest of this section.


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2 54 T H E R E V I E W O F E C O N O M IC S AND S TATI S TI C S

= v W ( t ) (46)where V = the marginal propensity to consumeout of wealth.

The tools of comparative statics are used toexam ine th e effect of shifts in the mea n an dvariance on consumption behavior in this mod-el. T he com parison is between two economieswith different investment opportunities, butwith the individuals in both economies havingthe same utility function.If 8 is a financial parameter, then define[ ]io, the p artial- derivative of consump--tion with respect to 8, I, [W,] being held fixed,as the intertemporal generalization of th eHicks-S lutsky "substitution" effect,for static models. [ a C * / a 8 - ( a C * / a e ) IO]will be defined as the intertemporal "income"or "wealth" effect. Then, from equation (22)with I, held fixed, one derives by tot al differen-tiation,

(47)From equations (24) an d (4 6) , b(0) = V-8,and so solving for (aWo /aO)io in (4 7) ) we canwrite i t as

Consider the case where 8 = a,, then from( 46) ,av (8-1)-= - 6 (49aa*and from (48) ,Thus, we can derive the substitution effect ofa n increase in the mean of the composite port-folio as follows,

Because a c * / a ~ * =a v / a ~ * ) W , = ( a- 1 ) /61W,, then t he income or wealth effect is

Th erefo re, by comb ining the effects of (5 1)and (52), one can see that individuals with lowrelative risk-aversion (0 < 6 < 1) will chooseto consume less now and save more to take ad-vantage of the higher yield available (i.e., thesubstitution effect dominates the income effect).For high risk-averters (6 > I) , the reverse istrue and the income effect dominates the sub-stituti on effect. I n the borderline case of Ber-noulli logarithmic utility (6 = I ) , the incomean d s ubs titutio n effect just offset one another.12I n a similar fashio n, consider the case ofe = - u.2,13 hen from, (46) and (4 8) ) we de-rive

and

effect. ( 5 4 )Further , aC*/a(-( r ,2) = (6 - 1)Wo/2, and so

6= - Wo > 0, the income effect.2 (5 5 )To compare the relative effect on consump-tion behavior of an upward shift in the meanversus a downward shift in variance, we ex-amine the elasticities. Define the elastic ity ofconsumption with respect to the mean as

and s imilarly, the elasticity of consumptionwith respect to the variance as,

For graphical simplicity, we plot el =[VEJa,] and e2=- VE 2/a,] and define k == M an y writers have independently discovered tha t Ber-noulli utility is a borderline case in various comparative-static situations. See, for example, Phelps [61 and ArrowD l.13Because increased variance for a fixed mean usually(always for normal variates) decreases the desirability ofinvestment for the risk-averter, it provides a more sym-me tric discussion to consider the effect of a decrease in

variance.


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L I F E T I M E P O R T F O L I O S E L E C T IO N 255higher future consumption while the high riskaverter will always choose to increase theamount of present consumption.

l and e2 are equal a t 8 = 1, 1/k . Th e< 1. > I ) , t h e( 6 > 1) will always increaseean. Because a high risk-averter prefersier flow of consum ption a t a lower level

< I ) , a low risk averter (0 < 8 < 1 ) wi llease in the mean than for the sam e

to accept a more er ratic flow of con-ption in retu rn for a higher level of con-If th ef the re tur ns is very small (i.e.,< I), then the high risk-averter will in-

ard sh ift in me an. Similarly, if the risk-> > 1) the low risk

> > I ) ) ,a decrease in risk willsions. W hen risk is unim porta nt (i.e.,< 1 ) ) they al l react stronger to an increased. Fo r all degrees of relativeness, the low risk -averter w ill give u p some

VIII Extension to Many AssetsThe model presented in section IV , can beextended to the m-asset case with little diffi-culty. Fo r simplicity, the solution is derived inthe infinite time horizon case, but the result issimilar for finite time. Assume the mth asset tobe the only certain asset with an instantaneousrate of return a, = r.14 Using the generalequations derived in section 11,and substi tut ing

nfor w,(t) = 1- 2, w,(t) where n= m - 1,i = lequations (6 ) an d (7 ) can be writ ten as,E ( t o ) [ W ( t ) - W(to) l= [wl ( tO)a-;) + Y ]W(to) h- C(t0)h + O(h2) ( 6 )an dE( to) [ ( W ( t ) - W(t0) )21

= w'(to) a to) W2(te)h+ O(h2) (7)wherewT(to)= a l ( t o ) , . . ,wn(to)], a n-vectorI -= al , . . . an]> = [ r , ..., ] a n-vector

n = odj], the n, X n variance-covariancematrix of the risky assetsn is symmetric and positive definite.Th en , the general form of (35 ) for m-assets is,in matrix notat ion,

0 = Max [ U ( C ) - p J (W ){ C P )+ J f ( W ) { [wf (a - j ) + r ] W - C )1+ JTT (W ) wW2] (58)an d instead of two, there will be m first-orderconditions corresponding to a maximization of(35) with respect to wl, . . ,w, and C. T h eoptimal decision rules corresponding to (42)and (43) in the two-asset case, are

''Clearly, if there were more than one certain asset, theone with the highest rate of return would dominate theothers.


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2 56 T H E R E V I E W O F E C O N O M IC S A N D S T A TI ST IC Sand

where w,*'(t) = [wl* ( t ) , . . . ,w,,* ( t ) 1.IX Constant Absolute Risk AversionSystem (*) of section 111,can be solved ex-plicitly for a second special class of utilityfunctions of th e form yielding consta nt absoluterisk-aversion. Let U(C) = -e-"/7, 7 > 0,where -U"(C) /Ur(C) = 7 is Pratt 's [17]me asure of absolute risk-aversion. Fo r con-venience, I return to the two-asset case and in-finite-time horizon form of system (*) whichcan be written in this case as,

I *(t) = - J r (W) (a - r ) /u2 W J"(W)subject to limit E[e -p t J (W ( t ) ) ] = 0t +m (19)whe re J ( W ) = ePtZ [ W ( t ) t] as defined in sec-tion VI.T o solve (17") of (*"), take a s a trial solu-tion,- -PJ ( W ) =- -qW.Q (61)By substitution of the trial solution into (17"),a necessary condition that S (W ) be a solutionto (17") is found to be tha t p and q must satisfythe following two algebraic equations:9 = rlf' (62)

p = e (63)The resulting optimal decision rules for port-folio selection and consumption a re,

(64)andw*(t) = (a - r )7 ru2W t)Comparing equations (64) and (65) with

their cou nterpa rts for the constant relative risk-aversion case, ( 42) and (4 3) , one finds tha tconsumption is no longer a constant proportionof wealth (i.e., marginal propensity to consumedoes not equal the average propensity) al-though it is still linear in wealth. Ins tea d of t heproportion of wealth invested in the risky assetbeing constant (i.e., w*(t) a constant), thetotal dollar value of wealth invested in therisky asset is kept constant (i.e., w* ( t ) W ( t )a constant). As one becomes wealthier, thepropo rtion of his wealth invested in the riskyasset falls, and asym ptotically, as W + m, oneinvests all his wealth in the certain asset andconsumes all his (c ertain) income. Althoughone can do the same type of comparative staticsfor this utility function as was done in sectionV II for the case of constant relative risk-aver-sion, it will not be done in this paper for thesak e of brevity a nd because I find this specialform of the utility function behaviorially lessplausible than constant relative risk aversion.It is interesting to note that the substitutioneffect in this case, [El is zero exceptae 1,'when r = t?

X Other Extensions of the ModelThe requirements for the general class ofprobability distributions which could be accept-able in this model are,(1) the stochastic process must be Markov-ian.(2 ) the first two moments of th e distributionmust be O ( A t ) a nd the higher-ordermoments o ( A t ) w here o ( . is the ordersymbol meaning "smaller order than."So, for exam ple, the simple Wiener process pos-tulated in this model could be generalized to

include ai = ai ( X I , . . . ,X,,W,t) and ui = u4( X I , . . . ,X,,W,t), where X i is the price of th eithasset. I n this case, there will be (m + 1)state variables and (17') will be generatedfrom the general Taylor series expansion ofZ[X2] , . . . ,X,,W,t] for ma ny variables. Aparticular example would be if the ith sset is abond which fluctuates in price for t < t,, butwill be called at a fixed price at time t = t,.Then a, = ai(Xi , t ) and ut = a i (X i 7 t ) > 0 whent < ti and u, = 0 for t > t,.A more general produc tion function of a neo-


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L I F E T I M E P O R T F O L I O S E L E C T IO N 257classical type could be introduced to replace [21 Dreyfus, S. E., Dynamic Programming and thethe sim d e linear one of this model. M irrlees Calczdz~s of Variations (New York : Academic[5 ] ha s examined this case in the context of agrowth model with Harrod-neutral technicalprogress a random variable. His equations(19) and (20) correspond to my equations(35) and (37 ) with the obvious proper substi-tutions for variables.Thus, the technique employed for this modelcan be extended to a wide class of economicmodels. However, because the optimality equa-tions involve a partial differential equation,computational solution of even a slightly gen-eralized model may be quite difficult.

R E F E R E N C E S[I] Arrow, K. J., "Aspects of the Theory of Risk-Bearing," Helsinki, Finland, Yrjo Jahnssonin Saa-tio, 1965.

Press , 1965).[3] Graaff, J. De V., "Mr. Harrod on Hump Saving ,"Economics (Feb. 1950), 81-90.

[4] I t6, K., "On Stochastic Differential Equations,"Memoirs , American Mathematical Society, No. 4(1965), 1-51.

[5] Mirrlees, J. A., "Optimum Accumulation UnderUncertainty," Dec. 1965, unpublished[6] Phelps, E. S., "The A ccumulation of Risky Capi-tal: A Sequential Utility Analysis," Econometrica,30 (1962), 729-743.['i] Pra tt, J., "Risk Aversion in the Small and in theLarge," Econom etrica, 32 (Jan. 196 4), 122-136.[8] Samuelson, P . A,, "Lifetime Portfolio Selection byDynamic Stochastic Programming," this REVIEW,L(Aug. 1969).[9] Tobin, J., "The Theory of Portfolio Selection,"The Theory of In teres t Rates , F. H. Hah n and

F. P. R. Brechling, (ed.) (London : MacM illanCo., 1965).

merton lifetime portfolio selection 1969

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