The Arithmetics of Par, Spot and Forward Curves

The understanding of the dynamics and shape of term structure of interest rates has been of a big interest to practitioners and academics. It is well documented that if the spot curve is monotonically upward-sloping (with decreasing growth rate) the par curve shows the same pattern, while the forward rate curve is first upward-sloping and then it inverts. Spot curve lies above the par curve, and the forward rate curve lies above the spot curve. If the spot curve is inverted, the pattern and ordering of the curves revert and the three curves are identical only if they are flat. This article aims to present mathematically the joint behavior of par, spot, and forward curves in discrete time setting