FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 77
Solution: First, i F 1 = iS 1 =4%. The rest of the calculation, using (3.6), is as
follows
i F 2 = (1 + iS 2 )2
1+i S − 1= (1.045)2 − 1=5.0024%,
1
1.04
and
i F 3 = (1 + iS 3 )3
(1 + i S − 1=(1.045)3
2 )2 (1.045) 2 − 1=4.5%,
i F 4 = (1 + iS 4 )4 (1.05)4
(1 + i S − 1=
3 )3 (1.045) 3 − 1=6.5144%.
Figure 3.2: Illustration of equation (3.5)
Forward rate
of interest
Spot rate
of interest
i F 1 i F 2
······
i F t
i S t
Time
0 1 2 ······
t − 1 t
Example 3.2: Suppose the forward rates of interest for investments in year 1, 2,
3 and 4 are, respectively, 4%, 4.8%, 4.8% and 5.2%. Calculate the spot rates of
interest for t =1,2,3and4.
Solution: First, i S 1 = iF 1
follows
=4%. From (3.5) the rest of the calculation is as
i S 2 = [ (1 + i F 1 )(1 + i F 2 ) ] 1
2
− 1= √ 1.04 × 1.048 − 1=4.3992%,
i S 3 = [ (1 + i F 1 )(1 + i F 2 )(1 + i F 3 ) ] 1
3
− 1=(1.04 × 1.048 × 1.048) 1 3 − 1
=4.5327%,
i S 4 = [ (1 + i F 1 )(1 + iF 2 )(1 + iF 3 )(1 + iF 4 )] 1
4
− 1,
=(1.04 × 1.048 × 1.048 × 1.052) 1 4 − 1,
=4.6991%.