JOURNAL OF ECONOMIC THEORY 3, 373-413(1971) Optimum Consumption and portfolio rules in a Continuous-Time modelk ROBERT C. MERTON Sloan School uf Manugement, Massachuset(s Institute of Technology Received September 30, 1970 A common hypothesis about the behavior of (limited liability )asset prices in perfect markets is the random walk of returns or(in its continous-time form)the"geometric Brownian motion"hypothesis which implies that asset prices are stationary and log-normally distributed A number of investigators of the behavior of stock and commodity prices have questioned the accuracy of the hypothesis. In particular, Cootner [2] and others have criticized the independent increments assump- tion, and Osborne [2] has examined the assumption of stationariness Mandelbrot [2] and Fama [2] argue that stock and commodity price changes follow a stable- Paretian distribution with infinite second moments The nonacademic literature on the stock market is also filled with theories of stock price patterns and trading rules to "beat the market, rules oftcn called"technical analysis "or"charting, and that presupposes a departure from random price changes In an earlier paper [12], I examined the continuous-time consumption- portfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the "geo- metric Brownian motion"hypothesis; i.e., I studied Max EJo U(C, t)de I would like to thank p. A samuelson. r. M. A Diamond.J. A. Mirrlees J. A. Flemming, and D. T. Scheffman for thei discussions Of errors are mine. Aid from the National Science Fo is gratefully acknowledged An earlier version of the paper was presented at the second World Congress of the Econometric Society, Cambridge, England. 1 For a number of interesting papers on the subject, see Cootner [2]. An excelle survey article is"Efficient Capital Markets: A Review of Theory and Empirical Work, by E, Fama, Journal of Finance, May, 1970
JOURNAL OF ECONOMIC THEDRY 3, 373-413 (1971) Optimum Consumption and Portfolio Rules in a Continuous-Time Model* ROBERT C. MERTQN Sloan School of Management, Massachusetts Institute of Techa?ology, Cambridge, Massachusetts 02139 Received September 30, 1970 1. INTRODUCTION A common hypothesis about the behavior of (limited liability) asset prices in perfect markets is the random walk of returns or (in its continous-time form) the “geometric Brownian motion” hypothesis which implies that asset prices are stationary and log-normally distributed. A number of investigators of the behavior of stock and commodity prices have questioned the accuracy of the hyp0thesis.l In particular, Cootner [2] and others have criticized the independent increments assumption, and Osborne [2] has examined the assumption of stationariness. Mandelbrot [2] and Fama [2] argue that stock and commodity price changes follow a stable-Paretian distribution with infinite second moments. The nonacademic literature on the stock market is also filled with theories of stock price patterns and trading rules to “‘beat the market,” rules often called “technical analysis” or “charting,” and that presupposes a departure from random price changes. In an earlier paper [12], I examined the continuous-time consumptionportfolio problem for an individual whose income is generated by capital gains on investments in assets with prices assumed to satisfy the “geemetric Brownian motion” hypothesis; i.e., I studied Max E jz eT(C, t) dt * I would like to thank P. A. Samuelson, R. M. Solow, P. A. Diamond, J. A. Mirrlees, J. A. Flemming, and D. T. Scheffman for their helpful discussions. Of course, a!1 errors are mine. Aid from the National Science Foundation is gratefully acknowledged. An earlier version of the paper was presented at the second World Congress of the Econometric Society, Cambridge, England. 1 For a number of interesting papers on the subject, see Cootner 121. An exceIIent survey article is “Efficient Capital Markets: A Review of Theory and Empirical Work,” by E. Fama, Journal of Finance, May, 1970. 373
374 MERTON where U is the instantaneous utility function, C is consumption, and e is the relative or constant absolute risk-aversion utility function, explicit solu- tions for the optimal consumption and portfolio rules were derived. The changes in these optimal rules with respect to shifts in various parameters such as expected return, interest rates, and risk were examined by the technique of comparative statics The present paper extends these results for more general utility functions, price behavior assumptions, and for income generated also from non capital gains sources. It is shown that if the"geometric Brownian motion hypothesis is accepted, then a general“ Separation'”or“ mutual fund theorem can be proved such that, in this model, the classical Tobin mean ariance rules hold without the objectionable assumptions of quadratic utility or of normality of distributions for prices. Hence, when asset prices are generated by a geometric Brownian motion, one can work with he two-asset case without loss of generality. If the further assumption is made that the utility function of the individual is a member of the family of utility functions called the"HARA"family, explicit solutions for the optimal consumption and portfolio rules are derived and a number of theorems proved. In the last parts of the paper, the effects on the consumption and portfolio rules of alternative asset price dynamics, in hich changes are neither stationary nor independent, are examined along with the effects of introducing wage income, uncertainty of life expectancy,and the possibility of default on(formerly)"risk-free 2. A DIGRESSION ON ITO PROCESSES To apply the dynamic programming technique in a continuous-time model, the state variable dynamics must be expressible as Marko tochastic processes defined over time intervals of length h, no matter how small h is. Such processes are referred to as infinitely divisible time. The two processes of this type? are: functions of Gauss-Wiener Brownian motions which are continuous in the"space"variables and functions of Poisson processes which are discrete in the space variables Because neither of these processes is differentiable in the usual sense, a more general type of differential equation must be developed to express the dynamics of such processes. a particular class of continuous-time 2 I ignore those infinitely divisible processes with infinite moments which include those members of the stable Paretian family other than the normal
374 MERTON where U is the instantaneous utility function, C is consumption, and E is the expectation operator. Under the additional assumption of a constant relative or constant absolute risk-aversion utility function, explicit solutions for the optimal consumption and portfolio rules were derived. The changes in these optimal rules with respect to shifts in various parameters such as expected return, interest rates, and risk were examined by the technique of comparative statics. The present paper extends these results for more general utility functions, price behavior assumptions, and for income generated also from noncapital gains sources. It is shown that if the “geometric Brownian motion” hypothesis is accepted, then a general ‘Separation” or “mutual fund” theorem can be proved such that, in this model, the classical Tobin meanvariance rules hold without the objectionable assumptions of quadratic utility or of normality of distributions for prices. Hence, when asset prices are generated by a geometric Brownian motion, one can work with the two-asset case without loss of generality. If the further assumption is made that the utility function of the individual is a member of the family of utility functions called the “HARA” family, explicit solutions for the optimal consumption and portfolio rules are derived and a number of theorems proved. In the last parts of the paper, the effects on the consumption and portfolio rules of alternative asset price dynamics, in which changes are neither stationary nor independent, are examined along with the effects of introducing wage income, uncertainty of life expectancy, and the possibility of default on (formerly) “risk-free” assets. 2. A DIGRESSION ON 1~6 PROCESSES To apply the dynamic programming technique in a continuous-time model, the state variable dynamics must be expressible as Markov stochastic processes defined over time intervals of length h, no matter how small h is. Such processes are referred to as infinitely divisible in time. The two processes of this type2 are: functions of Gauss-Wiener Brownian motions which are continuous in the “space” variables and functions of Poisson processes which are discrete in the space variables. Because neither of these processes is differentiable in the usual sense, a more general type of differential equation must be developed to express the dynamics of such processes. A particular class of continuous-time 2 I ignore those infinitely divisible processes with infinite moments which include those members of the stable Paretian family other than the normal
CONSUMPTION AND PORTFOLIO RULES 375 Markov processes of the first type called Ito Processes are defined as the solution to the stochastic differential equation dP=f(P, tdt+g(P, t)dz where P,, and g are n vectors and z(t)is an n vector of standard normal random variables. Then dz( t) is called a multidimensional wiener process (or Brownian motion).4 The fundamental tool for formal manipulation and solution of stochastic processes of the Ito type is Ito,'s Lemma stated as followss LEMMA. Let F(Pi,, Pn, t be a C2 function defined on R"XIO, co) and take the stochastic integrals Pi(t)=P:(O)+f(P, s)ds+ g(P, s)dzt, i=l,,; then the time-dependent random variable y= f is a stochastic integral and its stochastic diferential is d=∑F dp2++l∑∑ a"F dp, dPi where the product of the differentials dP, dP, are defined by the multi Pu dt, i, j dzi dt=0 Processes are a special case of a more general class of stochastic processes trong diffusion processes(see Kushner [9, p. 22].(1)is a short-hand expression P()=P(O)+f(P, s)ds+ g(P, s)dz, here P()is the solution to(1) with probability one Arigorous discussion of the meaning of a solution to equations like(1)is not presented here. Only those theorems needed for formal manipulation and solution of stochastic differential equations are in the text and these without proof. For a complete discussion of Ito Processes, see the seminal paper of Ito [7l, Ito and McKean [8], and McKear [11]. For a short description and some proofs, see Kushner 9, pp. 12-18]. For an euristic discussion of continuous-time Markov processes in general, see Cox and Miller [3, Chap. 51 4 dz is often referred to in the literature as"Gaussian White Noise. There are some imposed on the functions f and g. It is assumed throughout hat such conditions are satisfied. For the detail 5 See McKean [11, pp 32-35 and 44] for proofs of the Lemma in one and n dimen- sions
CONSUMPTION AND PORTFOLIO RULES 375 Markov processes of the first type called It6 Processes are defined as the solution to the stochastic differential equation3 dP = f (P, t) dt + g(P, t) dz, 02 where P, f, and g are n vectors and z(t) is an rz vector of standard normal random variables. Then dz(t) is called a multidimensional Wiener process (or Brownian motion).4 The fundamental tool for formal manipulation and solution of stochastic processes of the It6 type is Ita’s Lemma stated as follow@ LEMMA. Let F(Pl ,..., P, , t) be a C2 function dejned on and take the stochastic integmfs then the time-dependent random variable Y = F is a stochastic integral and its stochastic d@erential is where the product of the difSerentiaIs dPi dPj are defijled by the muktiplication rub dz, dzj = ,aij dt, i, j = I,..., n, dzi dt = 0, i = I,.~., y1 9 3 It6 Processes are a special ease of a more general cIass of stochastic processes called Strong diffusion processes (see Kushner 19, p. 221). (1) is a short-hand expression for the stochastic integral P(t) = P(0) + j-‘-f@‘, s) ds + jt g(P, s) dz, Ll 0 where P(t) is the solution to (1) with probability one. A rigorous discussion of the meaning of a solution to equations like (I) is not presented here. Only those theorems needed for formal manipulation and solution of stochastic differential equations are in the text and these without proof. For a complete discussion of Pto Processes, see the seminal paper of It6 173, It6 and McKean IS], and McKean [ll]. For a short description and some proofs, see Kushner [9, pp- 12-IS]. For an heuristic discussion of continuous-time Markov processes in general, see Cox and Miller [3, Chap. 51. * dz is often referred to in the literature as “Gaussian White Noise.” There are some regularity conditions imposed on the functions f and g. It is assumed throughout the paper that such conditions are satisfied. For the details, see [9] or [ll]. 5 See McKean [ll, pp. 32-35 and 441 for proofs of the Lemma in one and IZ dimensions
376 where pi is the instantaneous correlation coefficient between the wiener processes dz and dz,B Armed with Itos Lemma, we are now able to formally differentiate most smooth functions of Brownian motions (and hence integrate stochastic differential equations of the Ito typc). Before proceeding to the discussion of asset price behavior, another concept useful for working with Ito Processes is the differential generator (or weak infinitesimal operator) of the stochastic process P(t). Define the function G(P, t)by G,D=期E[+mhO=m2y hen the limit exists and where"Et' is the conditional expectation operator, conditional on knowing P(t). If the Pr(t) are generated by Ito Processes, then the differential generator of P, Lp, is defined by where f=(1…,mn),g=(g1…,gn),anda≡g;gPa, Further,,ican be shown that G(P, t)=SP[G(P, o) G can be interpreted as the"average"or expected time rate of change of on rule has dard nog nane variates (e.g, see [3D Warning: derivatives(and integrals) of functions of Brownian motions are similar to, but different from, the rules for deterministic differentials and integrals For example, P()=P(O)eian=P(O)e-(0)-3, then dP Pdz. Hence d≠log(P)Po) Stratonovich [15] has developed a symmetric definition of stochastic differential equations which formally follows the ordinary rules of differentiation and integration However this alternative to the ito formalism will not be discussed here
316 MERTON where pij is the instantaneous correlation coeficient between the Wiener processes dzi and dzj .6 Armed with Ito’s Lemma, we are now able to formally differentiate most smooth functions of Brownian motions (and hence integrate stochastic differential equations of the Iti, type).’ Before proceeding to the discussion of asset price behavior, another concept useful for working with It6 Processes is the differential generator (or weak infinitesimal operator) of the stochastic process P(t). Define the function @(P, t) by e(P, t) = iii Et [ GW + h), t + h) - G(W), t) h 1 9 (2) when the limit exists and where “E,” is the conditional expectation operator, conditional on knowing P(t). If the Pi(t) are generated by It6 Processes, then the differential generator of P, Zp , is defined by wheref = (fi ,..A, g = (8, ,..., g,), and aij = gigjpij . Further, it can be shown that G(P, t) = Z,[G(P, t)]. (4) C? can be interpreted as the “average” or expected time rate of change of 6 This multiplication rule has given rise to the formalism of writing the Wiener process differentials as dzi = Y* v’% where the z are standard normal variates (e.g., see [3]). ’ Warning: derivatives (and integrals) of functions of Brownian motions are similar to, but different from, the rules for deterministic differentials and integrals. For example, if p(i) = p(O) e&-tt = p(O) ez(t)-z(o)-*t, then dP = Pdz. Hence j$ j: dz # log U’(r)/P(O)) . Stratonovich 1151 has developed a symmetric definition of stochastic differential equations which formally follows the ordinary rules of differentiation and integration. However. this alternative to the It6 formalism will not be discussed here
CONSUMPTION AND PORTFOLIO RULES the function G(P, t)and as such is the natural generalization of the ordinary time derivative for deterministic functions 3. ASSET PRICE DYNAMICS AND THE BUDGET EqUATION e: Throughout the paper, it is assumed that all assets are of the limited ability type, that there exist continuously-trading perfect markets with no transactions costs for all assets, and that the prices per share, P(t)) are generated by Ito Processes, i.e. pi-a e, t)d e, t) dz where a is the instantaneous conditional expected percentage change in price per unit time and o 2 is the instantaneous conditional variance per unit time. In the particular case where the "geometric Brownian motion hypothesis is assumed to hold for asset prices, ai and o, will be constants For this case, prices will be stationarily and log-normally distributed and it will be shown that this assumption about asset prices simplifies the continuous-time model in the same way that the assumption of normality of prices simplifies the static one-period portfolio model c. To derive the correct budget equation, it is necessary to examine the rete-time formulation of the model and then to take limits carefully to obtain the continuous-time form. Consider a period model with periods of length h, where all income is generated by capital gains, and wealth w(t) and Pit) are known at the beginning of period t. Let the decision variables be indexed such that the indices coincide with the period in which the decisions are implemented. Namely, let M(t)= number of shares of asset i purchased during period t, i.e., between t and t+h C(t)=amount of consumption per unit time during period t. A heuristic method for finding the differential generator is to take the conditional xpectation of dG (found by Ito,s Lemma)and"divide"by dt. The result of this opera- tion will be LpIGl, i.e,, formally dt EdG)=G=sPIGJ The"p"operator is often called a Dynkin operator and is often written as"Dp 642/3/4-3
CONSUMPTION AND PORTFOLIO RULES 377 the function G(P, t) and as such is the natural generalization of the ordinary time derivative for deterministic functions.* 3. ASSET PRICE DYNAMICS AND THE BUDGET EQUATION Throughout the paper, it is assumed that all assets are of the limite liability type, that there exist continuously-trading perfect markets with no transactions costs for all assets, and that the prices per share, (p,(t)>, are generated by Ito Processes, i.e., dP. -2 D = q(P, t) dt + ai(P, t) dzizi , where 01~ is the instantaneous conditional expected percentage change in price per unit time and oi2 is the instantaneous conditional variance per unit time. In the particular case where the “geometric Brownian motion hypothesis is assumed to hold for asset prices, 01~ and gi will be constants. For this case, prices will be stationarily and log-normally distributed and it will be shown that this assumption about asset prices simplifies the continuous-time model in the same way that the assumption of normality of prices simplifies the static one-period portfolio model. To derive the correct budget equation, it is necessary to examine t discrete-time formulation of the model and then to take limits carefu to obtain the continuous-time form. Consider a period model with periods of length h, where all income is generated by capital gains, and wealth, W(t) and Pi(t) are known at the beginning of period t. Let the de variables be indexed such that the indices coincide with the per which the decisions are implemented. Namely, let iVi(t) = number of shares of asset i purchased during period t, i.e., between t and t + h and (61 C(l) = amount of consumption per unit time during period t. 8 A heuristic method for finding the differential generator is to take the conditional expectation of dG (found by ItUs Lemma) and “divide” by &. The result of this operation will be 2$[G], i.e., formally, ; &(dG) = d = 6pp[G]. The “2&” operator is often called a Dynkin operator and is often writterr as “DP”. 642/3/4-3
MERTON The model assumes that the individual"comes into"period t with wealth invested in assets so that ()=∑N(t-h)PO) Notice that it is N,t-h because N,(t- h) is the number of shares purchased for the portfolio in period(t-h) and it is Pi (t)because P:(t) is the current value of a share of the i-th asset. The amount of consumption for the period, C(t)h, and the new portfolio, N(t),are simultaneously chosen, and if it is assumed that all trades are made at(known) curr prices, then we have that C()h=∑[N()-N(t-h)P() The " dice''are rolled and a new set of prices is determined P,(t+h). and the value of the portfolio is now >i N(t) Pi(t+ h). So the individual comes into"period (t t h) with wealth w(t+h)=2i N(t)P(t+h) and the process continues Incrementing (7)and( 8)by h to eliminate backward differences, we C(t+ hh=IN(t+h)-N(o)]P(t+h) [Nt +h)-N(tIPit+h-PioI +∑[Nt+h)-NO)P(t) W(t+h)=∑N(t)Pt+h) (10) Taking the limits as h-0, we arrive at the continuous version of (9) and (10) C()dt=∑dN)dP(t)+∑dNt)Pt) 9 We use here the result that It& Proccsscs are right-contin P( and w() are right-continuous. It is assumed that ntinuous unction, and, throughout the paper, the choice of C(t)is ed class of
378 MERTON The model assumes that the individual “comes into” period t with wealth invested in assets so that W(t) = i Ni(t - h) P,(t). (7) 1 Notice that it is N,(t - h) because Ni(t - h) is the number of shares purchased for the portfolio in period (t - h) and it is Pi(t) because P,(t) is the current value of a share of the i-th asset. The amount of consumption for the period, C(t) h, and the new portfolio, N,(t), are simultaneously chosen, and if it is assumed that all trades are made at (known) current prices, then we have that -C(t) h = i [N,(t) - Ni(t - h)] P,(t). 1 (8) The “dice” are rolled and a new set of prices is determined, Pi(t + h), and the value of the portfolio is now C: Ni(t) Pi(t + h). So the individual “comes into” period (t + h) with wealth W(t + h) = Cf N,(t) Pi(t + h) and the process continues. Incrementing (7) and (8) by h to eliminate backward differences, we have that -c(t + h) h = i [Ni(t + h) - N,(t)1 I’& + h) 1 = 5 [Ni(t + h) - K(t)lV’i(t + 4 - f’,(t)1 + i IIN& + 4 - JJi(Ol Pi(t) 1 and W(t + h) = ‘f N,(t) Pi(t + h). 1 (9) Taking the limits as h + O,v we arrive at the continuous version of (9) and (lo), -C(t) dt = =f dNi(t) dP,(t) + i d&(t) Pi(t) (9’) 1 1 9 We use here the result that It6 Processes are right-continuous 19, p. 151 and hence P,(t) and w(t) are right-continuous. It is assumed that C(r) is a right-continuous function, and, throughout the paper, the choice of C(t) is restricted to this class of functions
CONSUMPTION AND PORTTOLIO RULES 379 W()=∑N()P(t) (10) Using Ito's Lemma, we differentiate(10) to get dH=2AdP2+∑NP+∑wN,dP1 The last two terms, >i dN Pi+li dn dPi, are the net value of additions to wealth from sources other than capital gains, 10 Hence, if dy(t)=(possi bly stochastic) instantaneous flow of noncapital gains (wage) income then we have that d-c)d-∑NP1+∑N,dP From(11)and (12), the budget or accumulation equation is written as dW=∑N(t)dP+d-C(t)d (13) It is advantageous to eliminate N,t) from(13)by defining a new vari w(t)=N(o)Po/w(n), the percentage of wealth invested in the i-th at time t. Substituting for dpi/Pi from(5), we can write( 13)as d=∑wWat-Cdt++∑wWo;dz where, by definition,∑w学1.1 Until Section 7, it will be assumed that dy =0, i.e., all income is derived from capital gains on assets. If one of the n-assets is"risk-free 10 This result follows directly from the discrete-time argument used to derive funds fro C(odt is replaced by a general do(t)where do(r)is the instantancous How of om all noncapital gains sources starting with the discrete-time formulation It is not from the continuous version directly whether dy-c()dt equais 11 There are no other restrictions on the individual w because borrowing and short
CONSUMPTION AND PORTFOLIO RULES 378 and W(t) = -f N,(t) P,(t). (IO’) 1 Using Ito’s Lemma, we differentiate (10’) to get The last two terms, C,” dNtPi + C:” dNi dP, , are the net value of additions to wealth from sources other than capital gains.lO IHence, if dy(t) = (possibly stochastic) instantaneous flow of noncapital gains (wage) income, then we have that dy - C(t) dt = i dN,P, +- f’ dNi dP, . 1 1 From (11) and (12), the budget or accumulation equation is written as dW = i N,(t) dP. 2 + dy - C(t) dt. 1 (13) It is advantageous to eliminate N,(t) from (13) by defining a new variable; ~~(1) = N,(t) P,(t)/ W(t), the percentage of wealth invested in the 6th asset at time f. Substituting for dPi/Pi from (5), we can write (13) as dW = f wi Woli dt - C dt + dy + i wi Woi dzi , (14) 1 1 where, by definition, CT uri 7 l.ll Until Section 7, it will be assumed that dy = 0, i.e., all income is derived from capital gains on assets. If one of the n-assets is “risk-free” I0 This result follows directly from the discrete-time argument used to derive (9’) where -C(t) dt is replaced by a general do(t) where &(t) is the instantaneous flow of funds from all noncapital gains sources. It was necessary to derive (12) by starting with the discrete-time formulation because it is not obvious from the continuous version directly whether dy - G(t)& equals C; dNtPi + Cy dNi dP, or just CT dNtP, . I1 There are no other restrictions on the individual wi because borrowing and shortselling are allowed
380 MERTON (by convention, the n-th asset), then on=0, the instantaneous rate of rcturn, an, will be called r, and(14)is rewritten as dW=∑w(4-r)W+(rW-C+d+∑Wod;,(14 where m=n-l and the wi, . Wm are unconstrained by virtue of the fact that the relation wn=1-2i w will ensure that the identity constraint in(14)is satisfied. 4. OPTIMAL PORTFOLIO AND CONSUMPTION RULES THE EQUATIONS OF OPTIMALITY The problem of choosing optimal portfolio and consumption rules for an individual who lives T years is formulated as follows max Eo U(C(), t)dt+B(W(T), T) subject to: w(O)= Wo: the budget constraint(14), which in the case of "risk-free"asset becomes(14); and where the utility function(during life)U is assumed to be strictly concave in C and the bequest "function B is assumed also to be concave in w.2 programming is used. Define J(W,P, t)=mxE U(C, s)ds+ B(W(T),T) where as before, "Et "is the conditional expectation operator, conditional nd Pa)=Pi. Define (w, C; W, P,t)=U(C, t)+9[] 12 Where there is no"risk-free" asset, it is assumed that no asse as a linear combination of the other assets, implying that the nn matrix of returns, s=oil, where ou E Pu; ; is nonsingular there is a""asset, the same assumption is made about the variance-covariance matrix
380 MERTON (by convention, the n-th asset), then (T, = 0, the instantaneous rate of return, E, , will be called r, and (14) is rewritten as dW = 2 wi(ai - r) W dt + (r W - C) dt + dy + f Wiai dzi , (14’) 1 1 where m = n - 1 and the wr ,..., w, are unconstrained by virtue of the fact that the relation w, = 1 - Cy wi will ensure that the identity constraint in (14) is satisfied. 4. OPTIMAL PORTFOLIO AND CONSUMPTION RULES: THE EQUATIONS OF OPTIMALITY The problem of choosing optimal portfolio and consumption rules for an individual who lives T years is formulated as follows: max E,, [ ,: u(C@>, t> dt + WV”), T,] (15) subject to: W(0) = W, ; the budget constraint (14), which in the case of a “risk-free” asset becomes (14’); and where the utility function (during life) U is assumed to be strictly concave in C and the “bequest” function B is assumed also to be concave in W.12 To derive the optimal rules, the technique of stochastic dynamic programming is used. Define J( W, P, t) = E Et [jr U(C, s> ds + WV), U] > (16) ,w where as before, “E,” is the conditional expectation operator, conditional on W(t) = Wand P,(t) = Pi . Define rb(w, c; w, P, t> = WC, 0 + aa (17) I2 Where there is no “risk-free” asset, it is assumed that no asset can be expressed as a linear combination of the other assets, implying that the n x it variance-covariance matrix of returns, 8 = [ud, where oij = pij~ioj, is nonsingular. In the case when there is a “risk-free” asset, the same assumption is made about the “reduced” m x m variance-covariance matrix
CONSUMPTION AND PORTFOLIO RULES given wi(t)=wi, C(t)=C, w(t=w, and Pi()=Pi. 3 From the theory of stochastic dynamic programming, the following theorem provides the method for deriving the optimal rules, C* and w*. THEOREM I. If the Pi(tare generated by a strong diffusion process, U is strictly concave in C, and B is concave in W, then there exists a set of optimal rules (controls), w* and C*, satisfying >iw % 1 and J(W, P,t)=B(W, T) and these controls satisfy 0= (C*,w*: W, P, t)>p(C, w; W, P, t) fort∈[0,7 From Theorem i we have that & O(C, w; W, P, 1 In the usual fashion of maximization under constraint, we define the Lagrangian,L≡中+A[1-∑ w, where A is the multiplier and find the extreme points from the first-order conditions 0=Lc(C*,w*)=Uc(C*,1)-Jw (19) 0- La, (C*,w*)--A+Jwa,W+Jww2oniw,*w? +∑JwoP2W,k=1 (20) 0=LA(C*,w*)=1-∑w* 13"p"is short for the rigorous ge,w, the dynkin operator over the variables P 2∑2w+2∑Pa 14 For an heuristic proof of this theorem and the derivation of the stochastic Bellman equation, see Dreyfus [4] and Merton [12]. For a rigorous proof and discussion of usher 19, Chap Iv, especially 7
CONSUMPTION AND PORTFOLIO RULES given -9vi(t) = wi , C(t) = C, W(t) = JV, and P,(t) = Pi .I3 From the theory of stochastic dynamic programming, the following theorem provides the method for deriving the optimal rules, C* and w*. ?hEOREM 1.14 If the P,(t) are generated by a strong d~~~s~on process, U is strictiy concave in C, and B is concave in W, then there exists a set of optimal rules (controls), w* and C*, satisfying Cy wi* = 1 and J(W, P, T) = B( W, T) and these controls satisfy 8 = +(c*, w”; w, P, t) 3 $(C, w; w7 P, t) for t E [O, T]. From Theorem I, we have that In the usual fashion of maximization under constraint, we define the kagrangian, L = $ + A[1 - Cl” wi] where h is the multiplier and 6find the extreme points from the first-order conditions 0 = L&C”, w*) = U,(C”, t) - Jw, (19) 0 = L,“,(C*, w*) = -A + JWZ~C W+ Jww f okjWj* W” 1 7z + c Jj wd’j W k = I,..., n, cm 0 = L,(C”, w*> = 1 - i wi*, o-1) 1 I3 “8” is short for the rigorous L$?$, the Dynkin operator over the variables P and W for a given set of controls w and C. I4 For an heuristic proof of this theorem and the derivation of the stochastic Bellman equation, see Dreyfus [4] and Merton [12]. For a rigorous proof and discussion of weaker conditions, see Kushner [9, Chap. IV, especially Theorem 71
382 MERTON where the notation for partial derivatives is Jw =aJ/aW,Jt=a/at, Uc=aU/aC,J=aJ/aP,, JM=0/aPi aP,, and Jsw= 02J/aP, aw Because Lcc= cc= UcC 0 (22) To solve explicitly for C* and w*, we solve the n +2 nondynamic implicit equations, ( 19H(21), for C*, and w*, and A as functions of Jw Jww, Jw, w, P, and t. Then, C* and w* are substituted in(18)which now becomes a sccond-ordcr partial differential equation for J, subject to the boundary condition J(W, P, T)=B(W, T). Having (in principle at least) solved this equation for J, we then substitute back into(19)(21) to derive the optimal rules as functions of w, P, and I. Define the inverse function G=[Uc]-. Then, from(19), (23) To solve for the wi * note that(20) is a linear system in wi* and hence can be solved explicitly, Define 22= [ou] n x n variance-covariance matrIx, Eliminating A from(20), the solution for wr* can be written as k*=h2(P,t)+m(P,W,t)8(P,t)1f(P,W,t),k=1,…,n,(25) where∑nh2=1,∑1gk=0,and∑1fk≡0.16 IS52- exists by the assumption on $2 in footnote 12. h (P,t)=eve/r; m(P, W, t)=-Jw/W/ww; ga(P,E ∑∑):P,W rP2-∑JmP∑n/rww
382 MERTON where the notation for partial derivatives is Jw SE aJ/a W, Jt = aJ/at, UC = aUjaC, Ji = aJ/aPi, Jij = a2J/aPi aPj, and Jjw = azJ/aPj a W. Because Lee = +cc = UC, c 0, -&ok = &to, = 0, -&ok = ~a2W2Jww, L %*j = 0, k fj, a sufficient condition for a unique interior maximum is that Jww 0. To solve explicitly for C* and w*, we solve the n + 2 nondynamic implicit equations, (19)-(21), for C*, and w*, and X as functions of Jw , J ww > Jiw , W, P, and t. Then, C* and w* are substituted in (18) which now becomes a second-order partial differential equation for J, subject to the boundary condition J(W, P, r) = B(W, T). Having (in principle at least) solved this equation for J, we then substitute back into (19)-(21) to derive the optimal rules as functions of W, P, and t. Define the inverse function G = [U&l. Then, from (19), C” = G(J, , t). (23) To solve for the wi*, note that (20) is a linear system in wi* and hence can be solved explicitly. Define 52 = [CT& the n x n variance-covariance matrix, [Vii] EE Q-l,15 (24) Eliminating X from (20), the solution for wk* can be written as wk* = h,(P, t) + m(P, K t> g,(P, t> +.ap, w, t), k = l,..., yt, (25) where C,” h, = 1, C: g, = 0, and C,“,fk E 0.16 I5 52-l exists by the assumption on 9 in footnote 12. n 16 h,(P, r> = c l+/r; m(P, w, t) = --Jw/WJwv ;
