An independent analysis of California’s three big pension funds has found a hidden shortfall of more than half a trillion dollars, several times the amount reported by the funds and more than six times the value of the state’s outstanding bonds.

The analysis was commissioned by Gov. Arnold Schwarzenegger, who has been pressing the State Legislature to focus on the rising cost of public pensions. Graduate students at Stanford applied fair-value accounting principles to California’s pension funds, using a method recently devised by two economists working in Illinois, Joshua D. Rauh of Northwestern University and Robert Novy-Marx of the University of Chicago.