Question 10.19: Can a forward rate be negative?

To see that the forward rates f(n,N) may be negative, let us analyse the case with n = 0 for simplicity. Then

f(0,N) = (N + 1)y(0,N + 1) − Ny(0,N)

and f(0,N) < 0 requires that (N + 1)y(0,N + 1) < Ny(0,N). The border case is when y(0,N + 1) = \frac{N}{N+1} y(0,N), which enables us to find a numerical example. For instance, for N = 8 and y(0, 8) = 9% a negative value f(0, 8) will be obtained if y(0, 9) < \frac{8}{9} × 9\% = 8\%.