FM for Actuaries

76 CHAPTER 3
side of equation (3.3) is less than the right-hand side, an arbitrageur will borrow 1
unit at time 0 over 2 years. 2 She will then lend 1 unit at time 0 over 1 year and at
the same time lend 1+i S 1 units, due at time 1 in the forward market (by entering
into such an agreement at time 0) over 1 year. This strategy will guarantee a profit
of (1 + i S 1 )(1 + iF 2 ) − (1 + iS 2 )2 > 0 without risk. On the other hand, if the lefthand
side of (3.3) is larger than the right-hand side, she will lend 2 years in the
spot market using capital borrowed for 1 year in the spot market with a forward
agreement to roll over the loan for 1 year at time 1. Thus, such arbitrage activities
will induce the two sides of equation (3.3) to be equal.
Equation (3.3) can be generalized to the following relationship concerning spot
and forward rates of interest
(1 + i S t ) t =(1+i S t−1) t−1 (1 + i F t ), (3.4)
for t = 2, 3, ··· . Readers are invited to formulate arbitrage arguments for the
proof of this equation. By repeatedly writing (1 + i S t−1 ) in terms of (1 + iS t−2 ) and
(1 + i F t−1 ), we can also conclude that
(1 + i S t )t =(1+i F 1 )(1 + iF 2 ) ···(1 + iF t ), (3.5)
which is illustrated in Figure 3.2.
Given i S t , the forward rates of interest i F t satisfying equations (3.4) and (3.5)
are called the implicit forward rates. Note that the quoted forward rates in the market
may differ from the implicit forward rates in practice, as when the market is
noncompetitive. Unless otherwise stated, however, we shall assume that equations
(3.4) and (3.5) hold, so that it is the implicit forward rates we are referring to in our
discussions.
Using equation (3.5) we can compute the spot rates given the forward rates. To
compute the forward rates from the spot rates, we derive, from equation (3.4),
i F t = (1 + iS t )t
(1 + i S − 1. (3.6)
t−1 )t−1
The following examples illustrate the applications of these results.
Example 3.1: Suppose the spot rates of interest for investment horizons of 1, 2, 3
and 4 years are, respectively, 4%, 4.5%, 4.5%, and 5%. Calculate the forward rates
of interest for t =1,2,3and4.
2 An arbitrageur is an investor who exploits mispricings in the financial markets by making simultaneous
offsetting trades aimed at creating riskless profits.