Merton considers a continuous time market in equilibrium. The state variable (X) follows a
Brownian motion: : dX = \mu dt + s dZ The investor maximizes his
Von Neumann–Morgenstern utility: :E_o \left\{\int_o^T U[C(t),t]dt + B[W(T),T] \right\} where T is the
time horizon and B[W(T),T] the utility from wealth (W). The investor has the following constraint on wealth (W). Let w_i be the weight invested in the asset i. Then: : W(t+dt) = [W(t) -C(t) dt]\sum_{i=0}^n w_i[1+ r_i(t+ dt)] where r_i is the return on asset i. The change in wealth is: : dW=-C(t)dt +[W(t)-C(t)dt]\sum w_i(t)r_i(t+dt) We can use
dynamic programming to solve the problem. For instance, if we consider a series of discrete time problems: :\max E_0 \left\{\sum_{t=0}^{T-dt}\int_t^{t+dt} U[C(s),s]ds + B[W(T),T] \right\} Then, a
Taylor expansion gives: : \int_t^{t+dt}U[C(s),s]ds= U[C(t),t]dt + \frac{1}{2} U_t [C(t^*),t^*]dt^2 \approx U[C(t),t]dt where t^* is a value between t and t+dt. Assuming that returns follow a
Brownian motion: : r_i(t+dt) = \alpha_i dt + \sigma_i dz_i with: : E(r_i) = \alpha_i dt \quad ;\quad E(r_i^2)=var(r_i)=\sigma_i^2dt \quad ;\quad cov(r_i,r_j) = \sigma_{ij}dt Then canceling out terms of second and higher order: : dW \approx [W(t) \sum w_i \alpha_i - C(t)]dt+W(t) \sum w_i \sigma_i dz_i Using
Bellman equation, we can restate the problem: : J(W,X,t) = max \; E_t\left\{\int_t^{t+dt} U[C(s),s]ds + J[W(t+dt),X(t+dt),t+dt]\right\} subject to the wealth constraint previously stated. Using
Ito's lemma we can rewrite: : dJ = J[W(t+dt),X(t+dt),t+dt]-J[W(t),X(t),t+dt]= J_t dt + J_W dW + J_X dX + \frac{1}{2}J_{XX} dX^2 + \frac{1}{2}J_{WW} dW^2 + J_{WX} dX dW and the expected value: : E_t J[W(t+dt),X(t+dt),t+dt]=J[W(t),X(t),t]+J_t dt + J_W E[dW]+ J_X E(dX) + \frac{1}{2} J_{XX} var(dX)+\frac{1}{2} J_{WW} var[dW] + J_{WX} cov(dX,dW) After some algebra , we have the following objective function: : max \left\{ U(C,t) + J_t + J_W W [\sum_{i=1}^n w_i(\alpha_i-r_f)+r_f] - J_WC + \frac{W^2}{2} J_{WW}\sum_{i=1}^n\sum_{j=1}^n w_i w_j \sigma_{ij} + J_X \mu + \frac{1}{2}J_{XX} s^2 + J_{WX} W \sum_{i=1}^n w_i \sigma_{iX} \right\} where r_f is the risk-free return. First order conditions are: : J_W(\alpha_i-r_f)+J_{WW}W \sum_{j=1}^n w^*_j \sigma_{ij} + J_{WX} \sigma_{iX}=0 \quad i=1,2,\ldots,n In matrix form, we have: : (\alpha - r_f {\mathbf 1}) = \frac{-J_{WW}}{J_W} \Omega w^* W + \frac{-J_{WX}}{J_W} cov_{rX} where \alpha is the vector of expected returns, \Omega the
covariance matrix of returns, {\mathbf 1} a unity vector cov_{rX} the covariance between returns and the state variable. The optimal weights are: : {\mathbf w^*} = \frac{-J_W}{J_{WW} W}\Omega^{-1}(\alpha - r_f {\mathbf 1}) - \frac{J_{WX}}{J_{WW}W}\Omega^{-1} cov_{rX} Notice that the intertemporal model provides the same weights of the
CAPM. Expected returns can be expressed as follows: : \alpha_i = r_f + \beta_{im} (\alpha_m - r_f) + \beta_{ih}(\alpha_h - r_f) where m is the
market portfolio and h a portfolio to hedge the state variable. ==See also==