FM for Actuaries

Bond Yields and the Term Structure 231
Going through a similar argument, we can write down the equation of value for
Bond 4 as
P 4 = (1+C 4 )P3 ∗ exp [−δ 4 (t 4 − t 3 )] + P 2 C 4 exp [−δ 3 (t ∗ 41 − t 2 )]
+ P3 ∗ C 4 exp [−δ 4 (t ∗ 42 − t 3)] ,
where 6 P ∗ 3 =exp[−δ 1 t 1 − δ 2 (t 2 − t 1 ) − δ 3 (t 3 − t 2 )] ,
so that
(1 + C 4 )exp[−δ 4 (t 4 − t 3 )] + C 4 exp [−δ 4 (t ∗ 42 − t 3 )]
= P 4 − P 2 C 4 exp [−δ 3 (t ∗ 41 − t 2)]
P3
∗ .
Thus, the right-hand side of the above equation can be computed, while the lefthand
side contains the unknown quantity δ 4 . The equation can be solved numerically
for the force of interest δ 4 in the period (t 3 ,t 4 ). Finally, using (7.14), we
obtain the continuously compounded spot interest rate as
⎧
δ 1 , for 0 <t<t 1 ,
⎪⎨
i S t =
⎪⎩
δ 1 t 1 + δ 2 (t − t 1 )
, for t 1 ≤ t<t 2 ,
t
δ 1 t 1 + δ 2 (t 2 − t 1 )+δ 3 (t − t 2 )
, for t 2 ≤ t<t 3 ,
t
δ 1 t 1 + δ 2 (t 2 − t 1 )+δ 3 (t 3 − t 2 )+δ 4 (t − t 3 )
, for t 3 ≤ t<t 4 .
t
The above example shows that the instantaneous forward rate of interest can be
computed successively using the bond data, although numerical methods may be
required in some circumstances. Using the estimated step function of instantaneous
forward rates, the spot-rate curve can be computed by equation (7.14). 7
6 Note that P ∗ 3 is the price of a unit par value zero-coupon bond maturing at time t 3.
7 Empirically it may occur that the forward rates of interest over some intervals are found to be
negative. Such anomaly is an indication of the existence of arbitrage opportunities in the market
and may be due to the poor quality of the data. Thus, some data cleaning may be required prior to
applying the Fama-Bliss method.