C2
Immunization and Liability Matching
Fund future liabilities with two bonds whose present value and duration match them, then test whether the surplus survives a parallel shift in rates.
Learning focus: matching duration protects against small parallel shifts, and extra convexity is what turns rate moves in either direction into a surplus.
Valuation
%
Liabilities
Funding bonds
%
%
Semiannual coupons, bought today and valued at the discount yield.
Liability present value
6,251,067
Liability duration
9.776
Macaulay 10.01 yrs
Short bond face
1,704,509
27.0% of value, worth 1,688,235
Long bond face
4,422,780
73.0% of value, worth 4,562,831
Convexity gap
57.74
assets ≥ liabilities: immunized
Worst surplus in the test
0
Surplus after an immediate parallel shift
With present value and duration matched, the surplus is zero before any shift and flat at it. Extra convexity bends it upward, so it grows whichever way rates move. Large or non-parallel moves, and the passage of time, still call for rebalancing.
Shift test
| Shift | Assets | Liabilities | Surplus |
|---|---|---|---|
| -300bp | 8,626,327 | 8,399,759 | 226,568 |
| -200bp | 7,698,219 | 7,608,257 | 89,962 |
| -150bp | 7,290,117 | 7,242,254 | 47,863 |
| -100bp | 6,914,823 | 6,894,693 | 20,129 |
| -50bp | 6,569,367 | 6,564,603 | 4,764 |
| -25bp | 6,406,980 | 6,405,821 | 1,159 |
| 0bp | 6,251,067 | 6,251,067 | 0 |
| +25bp | 6,101,331 | 6,100,233 | 1,098 |
| +50bp | 5,957,491 | 5,953,216 | 4,275 |
| +100bp | 5,686,441 | 5,670,230 | 16,211 |
| +150bp | 5,435,926 | 5,401,335 | 34,591 |
| +200bp | 5,204,141 | 5,145,796 | 58,345 |
| +300bp | 4,790,373 | 4,672,054 | 118,319 |