FM for Actuaries

78 CHAPTER 3
We have defined forward rates of interest that are applicable over a single period.
We now define the multi-period forward rate i F t,τ as the annualized rate of
interest applicable over τ periods from time t to t + τ, fort ≥ 1 and τ>0, with
the rate being determined at time 0. Figure 3.3 illustrates the definition of i F t,τ .
Figure 3.3: Illustration of equation (3.7)
One-period
forward rate
Multiple-period
forward rate
i F t+1 i F t+2
··· i F t+τ
i F t,τ
Time
0 ··· t t +1 t +2 ··· t + τ − 1 t + τ
Using arbitrage arguments involving rollover strategies, we can see that the
following no-arbitrage relationships hold. 3
(1 + i F t,τ )τ =(1+i F t+1 )(1 + iF t+2 ) ···(1 + iF t+τ ), for t ≥ 1, τ>0, (3.7)
and
(1 + i S t+τ )t+τ =(1+i S t )t (1 + i F t,τ )τ , for t ≥ 1, τ>0. (3.8)
Example 3.3: Based on the spot rates of interest in Example 3.1, calculate the
multi-period forward rates of interest i F 1,2 and iF 1,3 .
Solution: Using (3.7) we obtain
i F 1,2 =[(1+i F 2 )(1 + i F 3 )] 1 2 − 1=(1.050024 × 1.045)
1
2 − 1=4.7509%.
Similarly, we have
i F 1,3 =(1.050024 × 1.045 × 1.065144) 1 3 − 1=5.3355%.
We may also use (3.8) to compute the multi-period forward rates. Thus,
i F 1,2 = [ (1 + i
S
3 ) 3
1+i S 1
] 1 2
− 1=
[ (1.045)
3
1.04
] 1
2
− 1=4.7509%,
3 Note that i F t applies to the period t − 1 to t, whereas i F t,τ applies to the period t to t + τ. It
should be noted that i F t,1 = i F t+1.