Substituting for w* and C* in( 18), we arrive at the fundamental partial diflerential equation for J as a function of w, P, and t I[G,门+J4+J +∑JaP+ JPP+∑JwP JwP∑ wwE* ∑J甲P J(W,P,T)=B(W,T).f(26) solved, the solution J could be substituted into (23)and (25)to obtain C* and w* as functions of w. p. and t For the case where one of the assets is"risk-free, " the equations are somewhat simplified because the problem can be solved directly as an unconstrained maximum by eliminating wn as was done in(14). In this case, the optimal proportions in the risky assets are Pe k=1,,m.(27) The partial differential J corresp 0=U[G, T]+J+Jw[rw-G]+2JKaiPi ∑JwPA(x1-r) j∑∑(4-m)(x4-n)-2m2∑ JiwJ; P: subject to the boundary condition J(W, P, T)=B(W,T) Although(28)is a simplified version of (26), neither (26) nor(28)lend themselves to easy solution. The complexities of (26) and ( 28)are caused
CONSUMPTION AND PORTFOLIO RULES 383 Substituting for w* and C* in (18), we arrive at the fundamental partial differential equation for J as a function of W, 19, and t, 0 = U[G, t] f Jt + Jw [ " ='Jkiakw - G] vk,aka,r - i f f vkla 11 subject to the boundary condition J(W, P, T) = B(W, T). If (26) were solved, the solution J could be substituted into (23) and (25) to obtain C* and w* as functions of W, P, and t. For the case where one of the assets is “risk-free,” the equations are somewhat simplified because the problem can be solved directly as an unconstrained maximum by eliminating W, as was done in (14’). In this case, the optimal proportions in the risky assets are k = I,..., m. (27) The partial differential equation for J corresponding to (26) becomes 0 = U!G, T] + Jt + Jw[rW - G] + -f JioliPi 1 subject to the boundary condition J( W, P, T) = B( W, T), Although (28) is a simplified version of (26), neither (26) nor (28) lend themselves to easy solution. The complexities of (26) and (28) are caused
IERTON by the basic nonlinearity of the equations and the large number of state variables. Although there is little that can be done about the non- linearities, in some cases, it may be possible to reduce the number of state variables 5. LOG-NORMALITY OF PRICES AND THE CONTINUOUS-TIME ANALOG TO TOBIN-MARKOWITZ MEAN-VARIANCE ANALYSI When, for k=1, .,", aR and Ox are constants, the asset prices have ationary, log-normal distributions. In this case, J will be a function of w and t only and not P. Then(26) reduces to 0=U[G, t]+J++Jw ∑1W-G+2r jWW (29) From(25), the optimal portfolio rule becomes he+ m(w, ngu whcr∑hk=1and∑Bk=0 and h, and ge arc constants. From(30), the following“ separation”or‘ mutual fund'” theorem can THEOREM II. 7 Given n assets with prices Pi whose changes are log normally distributed, then (1)there exist a unique (up to a nonsingular transformation) pair of mutual funds "constructed from linear combinations of these assets such that, independent of preferences(i. e, the form of the utility function), wealth distribution, or time horizon, individuals will be indifferent between choosing from a linear combination of these two funds or a linear combination of the original n assets. (2) If P is the price per share of either fund, then P is log-normally distributed. Further, (3)if fund's value held in the k-th Ak= percentage of the other mutual fund's value held in the k-th asset, then one can find that A=h2+(=) See Cass and Stiglitz [] for a general discussion of Separation theorems. The only degenerate case is when all the assets are identically distributed (. e, symmetry in which case, only one mutual fund is needed
384 MERTON by the basic nonlinearity of the equations and the large number of state variables. Although there is little that can be ‘done about the nonlinearities, in some cases, it may be possible to reduce the number of state variables. 5. LOG-NORMALITY OF PRICES AND THE CONTINUOUS-TIME ANALOG TO TOBIN-MARKOWITZ MEAN-VARIANCE ANALYSIS When, for k = l,..., n, elk and gk are constants, the asset prices have stationary, log-normal distributions. In this case, J will be a function of W and t only and not P. Then (26) reduces to 0 = U[G, t] + Jt + Jw [ ” ‘;vxr”ic W - G] + q - &J’; [ $ $ (29) From (25), the optimal portfolio rule becomes wt* = hk + m(W, t> g, 9 (30) where CT hl, = 1 and C: g, = 0 and h, and g, are constants. From (30), the following “separation” or “mutual fund” theorem can be proved. THEOREM II.17 Given n assets with prices Pi whose changes are lognormally distributed, then (1) there exist a unique (up to a nonsingular transformation) pair of “mutualfunds” constructedfrom linear combinations of these assets such that, independent of preferences (i.e., the form of the utility function), wealth distribution, OP time horizon, individuals will be indifferent between choosing from a linear combination of these two funds or a linear combination of the original n assets. (2) If Pf is the price per share of either fund, then Pf is log-normally distributed. Further, (3) tf 6, = percentage of one mutual fund’s value held in the k-th asset and tf A, = percentage of the other mutual fund’s value held in the k-th asset, then one can find that 6,=hk+ug7)), V k = I ,..., n, I7 See Cass and Stiglitz [l] for a general discussion of Separation theorems. The only degenerate case is when all the assets are identically distributed (i.e., symmetry) in which case, only one mutual fund is needed
CONSUMPTION AND PORTFOLIO RULLS 385 and Ak =h, where v, n are arbitrary constantS(v+O) Proof. (1 )(30)is a parametric representation of a line in the hyper- plane defined by Ei W**= 1. 18 Hence, there exist two linearly independent vectors(namely, the vectors of asset proportions held by the two mutual funds) which form a basis for all optimal portfolios chosen by the indi- viduals. Therefore, each individual would be indifferent between choosing a linear combination of the mutual fund shares or a linear combination of the original n assets (2)Let V= NP= the total value of (either) fund where N= number of shares of the fund outstanding. Let N= number of d=∑ NE dPi+∑ Pk dM2+∑ dP: dNk N, dP,+ Pr dN,+ dP, dN, (31) Pk dNk+2dPk dNk- net inflow of funds from non-capital sources 32 net value of new shares issued P, dN,+ dN, dP, From(31)and (32), we have that ∑NdP2 By the definition of V and ur, (33)can be rewritten as P 1 P ∑gkdt+∑ Kor dzx 18 See [i.p
CONSUMPTION AND PORTFOLIO RULES 385 and k = l,..., n, where v, 77 are arbitrary constarits (v # O), Proof. (1) (30) is a parametric representation of a line in the hyperplane defined by C: I+* = 1. I8 Hence, there exist two linearly independent vectors (namely, the vectors of asset proportions held by the two mutual funds) which form a basis for all optimal portfolios chosen by the individuals. Therefore, each individual would be indifferent between choosing a linear combination of the mutual fund shares or a linear combination of the original n assets. (2) Let Y = NJ’, = the total value of (either) fund where Nf = number of shares of the fund outstanding. Let Nk = number sf shares of asset k held by the fund and pk = N,J’,/V = percentage of total value invested in the k-th asset. Then V = C,” NTCPI, and But dV = 5 Nk dPk, + f Pi, dN,C + 1 dP, div, 1 1 = Nf dP, + Pf dNf + dP, dNf . (31) f PI, dNk + i dP, dlv, = net inflow of funds from non-capital-gain 1 1 sources = net value of new shares issued 02) = P, dN, + dNf dP, . From (31) and (32), we have that N,dP, = iN,dP,. 1 By the definition of V and pIc, (33) can be rewritten as dP, _ n dPI, --Tpkpb Pf = f ,,&k dt + f /hka, dzk . 1 1 I8 See [I, p. 151
386 MERTON By Itos Lemma and (34), we have that P()=PO)exp1ak-4∑∑)t+∑po.35) So, P,(t)is log-normally distributed 3)Let a(w, t;U)=percentage of wealth invested in the first mutual fund by an individual with utility function U and wealth W at time t. Then, (I-a) must equal the percentage of wealth invested in the second mutual fund. Because the individual is indifferent between these asset holdings or an optimal portfolio chosen from the original n assets, it must be that Hk*=h2+m(W,1)gk=ak+(1-a)A,k=1,…,n.(36) All the solutions to the linear system(36) for all W, t, and U are of the h2+(=2 k=1 (37) a= vm(w,t)+n Note that ∑=2(-ys) ED For the case when one of the assets is"risk-free, there is a corollary to Theorem IL. Namely, COROLLARY. If one of the assets is"risk -free, then the proportions of ach asset held by the mutual funds are 8=72y-4=(=1)n- 1-∑A
386 MERTON By Ito’s Lemma and (34), we have that kak t + i pkak 1 So, Pf(t) is log-normally distributed. (3) Let a(IY, t; U) = percentage of wealth invested in the first mutual fund by an individual with utility function U and wealth W at time t. Then, (1 - a) must equal the percentage of wealth invested in the second mutual fund. Because the individual is indifferent between these asset holdings or an optimal portfolio chosen from the original IZ assets, it must be that ,@k* =hk + m( w, t) g, = aal, + (1 - a) A,, k = l,..., ~1. (36) All the solutions to the linear system (36) for all W, t, and 7J are of the form 6, = hk + 0 gk , k = l,..,, n, V Note that and h, = hi,-+& k = l,..., n, a = vmW, t> + rl, v # 0. ;xk=;(hk-$gk) = l. (37) Q.E.D. For the case when one of the assets is “risk-free,” there is a corollary to Theorem II. Namely, COROLLARY. If one of the assets is “risk-free,” then the proportions of each asset held by the mutual funds are 6,= l--86,, A, = 1 - 5 x,< . 1 1