Duration Gap on an Annuity Block
A life insurer's immediate-annuity block has a present value of liabilities of $95 million with a Macaulay duration of 12.0 years, measured at a flat 5% yield. Assets backing the block have a present value of $100 million and a Macaulay duration of 9.0 years. The investment committee will not sell the portfolio outright but will fund any purchase by selling a slice of existing assets, which have the portfolio's average duration. A 20-year zero-coupon bond is available, and the committee proposes buying $20 million of it. The CIO argues that because assets exceed liabilities by $5 million, the block is safe regardless of duration.
1. Compute the market value of the 20-year zero required to immunize the surplus against a parallel shift in yields, and evaluate the $20 million proposal.
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Target asset duration. Immunization on a surplus basis requires A x D_A = L x D_L:
D_A = 12.0 x 95 / 100 = 11.4.
Current asset duration is 9.0, a gap of 2.4 years.
Sizing the purchase. Buying x of the 20-year zero and selling x of existing assets leaves asset value unchanged at $100M and gives
D_A(x) = [9(100 - x) + 20x] / 100 = 9 + 0.11x.
Set 9 + 0.11x = 11.4 => x = $21.82M.
The $20M proposal delivers D_A = 9 + 2.2 = 11.2, leaving a dollar-duration gap of 1120 - 1140 = -20 ($M-years) - small, but not zero, and the sign matters: a negative gap means surplus falls when rates fall.
Note on leverage. The zero has duration 20 against a liability duration of 12, so closing the gap requires putting 21.8% of assets into the longest-dated instrument available. That is a large, concentrated key-rate exposure, which the parallel-shift framework conceals.
2. Evaluate the CIO's surplus-cushion argument quantitatively for a 100 basis point parallel fall in yields, with and without the purchase.
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First-order approximation. dS ~= -(A D_A - L D_L) dy.
No purchase: A D_A - L D_L = 900 - 1140 = -240. A 100bp fall (dy = -0.01) gives dS = -(-240)(-0.01) = -$2.40M, taking surplus from $5.0M to $2.6M. A 2% fall takes it to $0.2M; a fall of just over 2% wipes the block's surplus out entirely. The $5M cushion buys roughly 200bp of adverse movement, and nothing more.
With $20M of the zero: gap is -20, so a 100bp fall costs -$0.20M (surplus $4.8M), and a 100bp rise gains $0.20M.
At the fully immunized $21.82M: first-order change is zero.
Verdict. The CIO is confusing a stock cushion with an immunized position. The $5M is not a buffer against rate moves; it is the residual equity that a duration mismatch can consume. Note also that the $20M proposal leaves the surplus mildly long-rate-sensitive in the wrong direction: surplus falls when rates fall, which is exactly when a life insurer's annuity block is most likely to be stressed by disintermediation and reserve strengthening.
3. The yield curve has steepened twice in three years, both times with the 20-year rate up 60bp while the 5-year was unchanged. How should this change the recommendation?
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It breaks the parallel-shift premise. Duration immunization only neutralizes a single-factor shift. A 20-year-only rise of 60bp hits the zero (duration 20, roughly 21.8% of assets) far harder than it hits the annuity liabilities (duration 12, spread across many payment dates):
- Asset value change from the 20-year move ~ -0.218 x 20 x 0.006 x 100 = -$2.62M
- Liability value change, at a key-rate duration of perhaps 1.5-1.8 years at the 20-year point ~ -$0.9M to -$1.1M
Net hit to surplus of roughly $1.5M against a $5M cushion - from a move that leaves the 5-year untouched and that a duration-matched portfolio would report as fully hedged.
Recommendation.
- Manage the block on key-rate durations, or better, on cash flows, not on a single Macaulay duration. The annuity payments are contractually fixed and non-callable, which makes a cash-flow match feasible and strictly superior to duration matching here.
- If cash-flow matching is impractical, cap the 20-year zero at the liability's own 20-year key-rate exposure and close the remaining gap with intermediate maturities, accepting a small duration mismatch in exchange for a much smaller key-rate mismatch.
- Report both a parallel-shift and a steepening scenario to the committee; the second has already happened twice and should not be treated as a stress case.