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《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 < z} = P(z). z- (8) Hence, Zt+1 - 1 is the percentage "yield" of each outcome. The most general probability distribution is admissible: i.e., a probability density over continuous z's, or finite positive probabilities at discrete values of z. Also I shall usually assume independence between yields at different times so that P(zo, z1, ... , Z ... * , ZT) = P(zt)P(Z1) ... P (zT) For simplicity, the reader might care to deal with the easy case Prob {Z = X} = 1/2 =Prob{Z=X-}, x> 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

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