Patent Document ID: 20120116988
Application ID: 12927175
Patent Flag: 0

Claim One:
1. A method for solving an expected utility maximization problem when asset returns are represented as a factor model, comprising the steps of: calculating, using a computer, an expected utility of the sum of a discrete and a continuous random variable, distributed independently, expressing the expected utility maximization problem as 
 max Eu (1+( R F ω +ε) T x ) 
 Ax=b,l≦x≦h; calculating, using the computer, the expected utility for given values of x based on a selected factor model estimation according to Eu ( ( 1 + R F ω + ε ) T x )  x = 1  Ω  ∑ ω ∈ Ω ∫ - τ + τ u ( 1 + R F ω T x + v x ) p ( v x )  v x where, given the independence of the ε i 's, 
 Γ x =ε T x=N (0,σ x 2 ) is normal distributed with mean zero and variance σ x 2 = ∑ i = 1 n σ i 2 x i 2 and therefore Eu ( ( 1 + R F ω + ε ) T x )  x = 1  Ω  ∑ ω ∈ Ω 1 2 π ∫ - τ + τ u ( 1 + R F ω T x + σ x v )  - v 2 2  v and for given value of x, calculating 1 2 π ∫ - τ + τ u ( 1 + R F ω T x + σ x v )  - v 2 2  v comprises integration of the utility function over a normally distributed random variable with mean value μ x ω =1+R F ωT x and variance σ x 2 =Σ i=1 n σ i 2 x x 2 , where the integration is only one-dimensional and thus can be carried out numerically using a trapezoidal method; calculating, using the computer, gradients of the objective at given value x with respect to x i as ∂ ∂ x i Eu ( 1 + ( R F ω + ε ) T x )  x = 1  Ω  ∑ ω ∈ Ω ∂ u _ x ω ∂ x i where ∂ u _ x ω ∂ x i = ∂ u _ x ω ∂ μ x ω R Fi ω + ∂ u _ x ω ∂ σ x 2 = 2 σ i 2 x i , where the integrations ∂ u _ x ω ∂ μ x ω = 1 2 π ∫ - τ + τ u ′ ( μ x ω + σ x v )  - v 2 2  v ∂ u _ x ω ∂ σ x 2 = 1 2 2 π σ x ∫ - τ + τ u ′ ( μ x ω + σ x v ) v  - v 2 2  v , are only one-dimensional and are carried out numerically using the trapezoidal method; and solving, using the computer, the expected utility maximization problem using gradient-based nonlinear programming; whereby the method takes advantage of the structure of the problem and requires on each iteration of the gradient-based nonlinear programming the numerical integration in one dimension only.