FM for Actuaries

3
Spot
Rates, Forward Rates
and the Term Structure
The annuity formulas in Chapter 2 are derived based on the assumption
that the rates of interest are constant through time and do not
vary with the periods of the payments due. In practice, however, we
would expect interest rates to change over time. Furthermore, the
rates of interest may vary according to the time to maturity (the due
date) of the payments. In this chapter we introduce the spot rate of interest
and discuss the relationship between the rates of interest and
the time to maturity, called the term structure of interest rates. We
also define the notion of the forward rate of interest, and describe the
relationship between the spot and forward rates of interest.
We discuss the relationship between the present and future values
assuming future payments earn the forward rates of interest. We
show the linkage between the accumulation function, and the spot
and forward rates of interest. The definition of the accumulation function
is extended so that it can be used to represent the term structure
of the forward rate of interest.
An interest rate swap is an agreement between two parties to exchange
cash flows based on interest rate movements in the future. It
can be used to convert a floating rate loan to a fixed rate debt or vice
versa. We discuss the determination of the swap rate and market
value of the swap agreement prior to its termination.
This chapter lays the background for term structure modeling. The
estimation of the yield curve and the theories for the determination of
the term structure of interest rates will be discussed in Chapter 7.