FM for Actuaries

268 CHAPTER 8
The Macaulay duration of the assets and liabilities can be calculated using equation
(8.3) to give
D A = 1 × 154.16 (1.1)−1 +3× 2,186.04 (1.1) −3 +5× 660.18 (1.1) −5
2,192.47
= 3.2461 years,
D L = 2 × 1,000 (1.1)−2 +4× 2,000 (1.1) −4
2,192.47
= 3.2461 years.
Furthermore, using equation (8.16), we get
C A = 2×1×154.16 (1.1)−1 +4×3×2,186.04 (1.1) −3 +6×5×660.18 (1.1) −5
(1.1) 2 × 2,192.47
=12.1704,
C L = 3 × 2 × 1,000 (1.1)−2 +5× 4 × 2,000 (1.1) −4
(1.1) 2 × 2,192.47
= 12.1676.
Since V A = V L , D A = D L and C A >C L , the conditions of the Redington immunization
strategy are met for Alfred’s strategy.
For (b), when there is an immediate one-time shift in interest rate from 10% to
9%, using equation (8.2), we have
V A = 154.16 (1.09) −1 +2,186.04 (1.09) −3 + 660.18 (1.09) −5 =$2,258.53,
V L = 1,000 (1.09) −2 +2,000 (1.09) −4 =$2,258.53,
S = 2,258.53 − 2,258.53 = 0.
We repeat the above calculations for interest rate of 11%, 15%, 30% and 80%. The
results are summarized as follows:
Table 8.4: Results of Example 8.11
i V A V L S
0.09 2,258.53 2,258.53 0.00
0.10 2,192.47 2,192.47 0.00
0.11 2,129.08 2,129.08 0.00
0.15 1,899.64 1,899.65 −0.02
0.30 1,291.40 1,291.97 −0.57
0.80 495.42 499.16 −3.74