FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 79
and similarly,
[ (1 + i
i F S
1,3 = 4 ) 4 ] 1
1+i S 1
3
− 1=
[ (1.05)
4
1.04
] 1
3
− 1=5.3355%.
3.2 The Term Structure of Interest Rates
Empirically the term structure can take various shapes, and its determination and
dynamic evolution has been one of the most widely researched topics in finance. A
sample of four yield curves of the US market are presented in Figure 3.4, and they
illustrate some of the most commonly observed term structures in history. 4
It can be seen that the spot-rate curve on 1998/06/30 is generally flat. The forward
rate is also flat at the short end of the time to maturity and overlaps with the
spot rate. Beyond maturity of 10 years, the spot rate increases gradually, and the
forward rate starts to exceed the spot rate. This is an example of a nearly flat term
structure. On 2000/03/31 we have a downward sloping term structure, where
the long-term spot rate is lower than the short-term spot rate. In this case the forward
rate drops below the spot rate, although it starts to climb up when the slope
of the spot rate is getting flatter beyond maturity of 10 years. In contrast, we have
an upward sloping term structure on 2001/08/31. Indeed, the spot rate increases
quite sharply at the short end of the yield curve, although its slope is gradually flattening
out as the time to maturity increases. Unlike the case of a downward sloping
yield curve, the forward rate exceeds the spot rate when the yield curve is upward
sloping. An upward sloping yield curve is also said to have a normal term structure
as this is the most commonly observed term structure empirically. Finally,
we have an inverted humped yield curve on 2007/12/31. The spot rate drops at
the short end of the maturity before it starts to increase at around maturity of 3
years. The forward-rate curve crosses the spot-rate curve from below and varies
quite a lot with the time to maturity.
Some questions may arise from a cursory examination of this sample of yield
curves. For example: How are the yield curves obtained empirically? What determines
the shape of the term structure? Why are upward sloping yield curves
observed more often? Does the term structure have any useful information about
the real economy? We shall answer some of these questions later in this book. In
particular, we shall discuss some methods for the estimation of the term structure.
In addition, we shall summarize some theories concerning the determination of the
term structure.
4 These yield curves are estimated by J. H. McCulloch and can be downloaded from
http://www.econ.ohio-state.edu/jhm/ts/ts.html.