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, LCw,- pCw, =0, L MpmR=.wWw Lu,w,=0, k*j, a sufficient condition for a unique interior maximum is that Jww<O(i.e,, that J be strictly concave in W). That assumed, as in immediate consequence of differentiating (19) totally with respect to W, we have >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 (i.e., that J be strictly concave in W). That assumed, as an immediate consequence of differentiating (19) totally with respect to W, we have ac* aw > 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 ;