Rate Relationships
Overview
Continuous Compounding
{% (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) %}
