Patent Document ID: 20080010181
Application ID: 11804692
Patent Flag: 0

Claim One:
1. A method for dynamic asset allocation system for portfolio optimization, comprising the steps of: representing a utility function as a piecewise exponential function with K pieces, where each piece represents a certain absolute risk aversion α i , where i=1,. .. , K, using the exponential function. Ŵ i ≦W≦Ŵ i+1 , u i (W i )=a i −b i exp(−α i W i ), where Ŵ i , i=1,. .. , K, be discrete wealth levels representing the borders of each piece i, such that below each Ŵ i the risk aversion is α i and above Ŵ i (till Ŵ i+1 ) the risk aversion is α i+1 , for all i=1,. .. , K; Computing absolute risk aversions α i that represent the desired function of risk aversion versus wealth by determining the coefficients of the exponential functions of each piece i by matching function values and first derivatives at the intersections Ŵ i , where the first derivative with respect to wealth is u i ′(W i )=b i α i exp(−α i W i ) and at each wealth level Ŵ i , representing the border between risk aversion α i and α i+1 , we obtain the following two equations 
 a i −b i e −α i Ŵ i =a i+1 −b i+1 e −α i+1 Ŵ i 
 b i α i e −α i Ŵ i =b i+1 α i+1 e −α i+1 Ŵ i , from which we calculate the coefficients a i+1 and b i+1 as b i + 1 = b i ⁢ α i α i + 1 ⁢ ⅇ ( α i + 1 - α 1 ) ⁢ W ^ i a i + 1 = a i - b i ⁡ ( 1 - α i α i + 1 ) ⁢ ⅇ - α i ⁢ W ^ i , where we set arbitrarily a 1 =0 and b 1 =1 and the piecewise exponential function may span the whole range of attainable wealth levels, where starting from parameters a 1 =0 and b 1 =1 and given risk versions α i , we compute all parameters a i+1 and b i+1 for each i=1,. .. K.