Let Ì¢è '_z be the Cauchy-Riemann operator and f be a real-valued Cn function in a neighborhood of the origin in the complex plane for which the nth order Cauchy-Riemann derivative of f is non-zero for all z other than 0 in that neighborhood. CarathÌÄå©odory conjectured that a sphere immersed in R3 must have at least two umbilics. This later motivated Loewner to conjecture an upper bound of n for the index of the nth order Cauchy-Riemann derivative of f as a vector fields. Both of these conjectures remain open. Recent work of F. Xavier produced a formula for computing the index of such vector fields in the case n = 2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for the nth order Cauchy-Riemann derivative that is valid for all n Ì¢'¡å´ 2. Structurally, our index formula provides a defect term for Loewner's conjecture containing geometric data extracted from Hessian-like objects associated with higher order derivatives of f. In particular, our index formula computes the Fredholm index of the Toeplitz operator on the unit circle whose symbol is the nth order Cauchy-Riemann derivative of f.