The term structure of interest rates and related functions

A t-year zero-coupon bond with a face value of 1 USD is a security that pays 1 USD at maturity, that is, in t years time. Let denote its market value, which is also called the t-year discount factor. The function is called the discount function. Based on the no-arbitrage assumption, it is usually assumed that , is monotonically decreasing, and that . It is also usually assumed that is twice continuously differentiable.

Let denote the continuously compounded annual return of the t-year zero coupon bond; it shall be defined as:

The function is called the (zero coupon) yield curve.

Let denote the instantaneous forward rate curve or simply the forward rate curve, where:

Here is the interest rate agreed upon by two parties in a hypothetical forward loan agreement, in which one of the parties commits to lend an amount to the other party in t years time for a very short term and at an interest rate that is fixed when the contract is...