A HOMOGENOUS DISTRIBUTION PROBLEM WITH APPLICATIONS TO FINANCE

THE PROBLEM OF DETERMINING THE CUMULATIVE DISTRIBUTION FUNCTION AND/OR MOMENTS OF THE OPTIMAL SOLUTION VALUE OF A NONLINEAR PROGRAM DEPENDENT UPON A SINGLE RANDOM VARIABLE IS CONSIDERED. THE PROBLEM IS DIFFICULT COMPUTATIONALLY BECAUSE ONE MUST IN EFFECT DETERMINE THE OPTIMAL SOLUTION TO AN INFINITE NUMBER OF NONLINEAR PROGRAMS. AN EXTREMELY SIMPLE ALGORITHM IS PRESENTED TO SOLVE THE PROBLEM IN THE SPECIAL CASE WHEN ALL FUNCTIONS IN THE NONLINEAR PROGRAM ARE HOMOGENOUS. IN THIS INSTANCE THE INFINITE CLASS OF OPTIMAL SOLUTIONS ARE KNOWN LINEAR TRANSFORMATIONS OF THE OPTIMAL SOLUTION TO A SINGLE NONLINEAR PROGRAM. THE DISTRIBUTION FUNCTION MAY THEN BE DETERMINED BY SUBSTITUTION OF AN EASILY CALCULATED VARIABLE INTO THE DISTRIBUTION FUNCTION OF THE RANDOM VARIABLE. THE RESULTS ARE USEFUL IN THE SOLUTION AND ANALYSIS OF A NUMBER OF FINANCIAL OPTIMIZATION PROBLEMS. PROBLEMS FROM THE ANALYSIS OF OPTIMAL CAPITAL ACCUMULATION AND PORTFOLIO SEPARATION ARE TREATED IN SOME DETAIL. EQUATIONS. TABLE. GRAPH. REFERENCES.