The frustration index of a signed graph is the smallest number of edges whose sign reversal yields a balanced signed graph (i.e., where all circles have positive sign product). In 1966, Petersdorf showed that the maximum frustration of a signed complete graph K_n is equal to floor[(n-1)²/4], and it is achieved by the all-negative signing. This is the only interesting graph family for which the problem has been solved. Using convex geometry, I show that the maximum frustration for a signed K_l,r is bounded above by

    r · (l/2) · [ 1 − 2^−(l−1) binom( l−1, floor[(l−1)/2] ) ]

and that this bound is achieved by a signing of K_l,2^l−1.