Overview
There are several definitions used in the description of an interest rate curve. This seciont summarizes some of the definitions and derives the relationships between them.
Continuous Compounding
Coninuous compounding results from the limit of compounding on a given period, as the length of that period goes to zero. Specifically,
{% (1 + \frac{R}{m})^m \rightarrow e^R %}
Calculations
The price of a 1 dollar zero coupon bond, received at T is
{% P(t,T) %}
Assumptions
- {% P(T,T) %} = 1
- {% P(t,T) %} is a differentiable function of T
The forward rate for time period [T,S]. at time t is defined as
{% F(t,T,S) = \frac{1}{S-T} (\frac{P(t,T)}{P(t,S)} - 1) %}
which means that we agree at time t to invest 1 dollar at time T, we receive
{% exp[(S-T)F(t,T,S)] %}
at time {% S %}
The Spot Rate
The spot rate for time period {% [t,T] %} is
{% F(t,T) = F(t,t,T) = \frac{1}{T-t} (\frac{1}{P(t,T)} - 1) %}
The continuously compounded forward rate if
{% R(t,T,S) = - \frac{ln P(t,T)}{T-t} %}
The continously compounded spot rate for [t,T]
{% R(t,T) = R(t,t,T) = - \frac{ln P(t,S) - ln P(t,T)}{S-T} %}
the instantaneous spot rate is
{% r(t) = \lim T \rightarrow t R(t,T) %}