FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 75
We now define i F t as the rate of interest applicable to the period t−1 to t, called
the forward rate of interest. Unlike i S t , which applies to t periods (from time 0 to t),
i F t applies to only one period (from time t − 1 to t). Also, this rate is determined at
time 0, although the payment is due at time t−1 (thus, the use of the term forward).
By convention, we have i S 1 ≡ iF 1 .However,iS t and i F t are generally different for
t =2, 3, ···. Figure 3.1 illustrates the definitions of i S t and i F t .
Figure 3.1:
Spot and forward rates of interest
i F 2
i F t
i S 2
Rates of
interest
i S 1
i S t
Time
0 1 2 ······
t − 1 t
Aplotofi S t against t is called the yield curve, and the mathematical relationship
between i S t and t is called the term structure of interest rates. While we
have defined two sets of interest rates, the spot and forward rates, they are not free
to vary independently of each other. To illustrate this point, we consider the case of
t =2. If an investor invests a unit amount at time 0 over 2 periods, the investment
will accumulate to (1 + i S 2 )2 at time 2. Alternatively, she can invest a unit payment
at time 0 over 1 period, and enter into a forward agreement to invest 1+i S 1 unit
at time 1 to earn the forward rate of i F 2 for 1 period. This rollover strategy will
accumulate to (1+i S 1 )(1+iF 2 ) at time 2. We shall argue that the two strategies will
accumulate to the same amount at time 2, so that
(1 + i S 2 )2 =(1+i S 1 )(1 + iF 2 ). (3.3)
The above equation holds if the capital market is perfectly competitive, sothat
no arbitrage opportunities exist. An assumption for a perfectly competitive capital
market is that the lending and borrowing rates are the same. 1 Thus, if the left-hand
1 This is an idealized condition to facilitate the proof. Equation (3.3) holds approximately when
the capital market is nearly perfectly competitive.