IBBC CAPITAL ACADEMY / 2027
Hedge liabilities, not just bond-price volatility
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Learning goals
- Diagnose the case using the stated constraints and quantitative evidence.
- Defend a conditional decision and identify what further evidence could reverse it.
Draft · approximately 95 minutes, excluding further revision.
Decision case · fictional facts
An institution owes a fixed 100 m cash payment in five years. The discount rate is 4% with annual compounding; ignore default and taxes. The liability has Macaulay duration 5 and modified duration 5/1.04 = 4.807692. A candidate asset portfolio has PV 82.19 m and the same modified duration. A broader bond index has modified duration 7 and uneven issuer exposure.
Single and multiple obligations
A maturity-matched default-free zero-coupon asset can directly fund a single nominal liability under compatible currency and settlement assumptions. Duration matching also needs matching present values and consideration of convexity, reinvestment and curve shifts. Multiple liabilities require cash-flow matching, duration and key-rate methods or combinations, with rebalancing as time and yields change.
Model the surplus under shocks
The relevant risk is assets minus liabilities, not asset volatility alone. A dollar-duration match offsets a small parallel yield change but leaves nonparallel shifts, credit spread, inflation and model risk. Derivative overlays can change duration efficiently but require collateral, counterparty and basis controls; they do not create the cash to meet the obligation.
Select a benchmark that matches purpose
Bond indexes can be difficult to replicate because of many issues, turnover, illiquidity, pricing differences and issuance weighting. Sampling, pooled vehicles and derivatives can approximate exposure. A benchmark should be specified in advance, measurable, appropriate and consistent with the investor’s liability objective; the broad index is not automatically suitable merely because it is familiar.
Bond indexes pose structural replication challenges
A bond index can contain thousands of issues with infrequent trades, estimated prices and rapid maturity or issuance changes. Debt weighting can increase exposure to large borrowers; it does not necessarily reward credit quality. Pricing at index valuation time can differ from executable dealer prices, creating tracking risk even when holdings appear close.
Passive exposure can be sampled or synthesized
A sampled bond portfolio can target duration, key rates, sector and credit exposures using fewer securities. A pooled fund centralizes implementation; futures or swaps can provide broad rate or index exposure with collateral, basis and counterparty demands. Compare net tracking, liquidity and operational complexity rather than assuming one method dominates.
Differentiate duration matching from exact cash-flow matching
Use the original fixed 100 m payment in five years at a 4% annual-compounded discount rate. The zero-coupon liability has Macaulay duration 5 years and modified duration 5/1.04. For this extension, compare an asset portfolio with the same PV and modified duration but different cash-flow dates. Approximate the liability change for a +100 bp yield move and then calculate the exact repriced liability.
A futures overlay can adjust first-order rate exposure but does not deliver the dated 100 m cash payment by itself. Explain how collateral liquidity, curve shape, asset default and reinvestment between cash-flow dates remain relevant after matching one duration number.
Model reasoning, sensitivity and self-check
Initial PV = 100/1.04^5 = 82.192711 m. Modified duration = 4.807692. First-order change = −82.192711×4.807692×0.01 = −3.951573 m, or −4.807692%. Exact PV at 5% = 100/1.05^5 = 78.352617 m: an exact change of −3.840094 m, or −4.672062%. The gap reflects curvature, not an arithmetic license to substitute Macaulay duration into a modified-duration formula.
For this annual zero, convexity = 5×6/1.04² = 27.736686. Adding 0.5×convexity×0.01² to the percentage approximation gives approximately −4.669009%, closer to exact repricing. A duration-matched coupon portfolio with different convexity can leave residual surplus volatility even for a parallel shift.
Self-check: name the duration measure, compounding convention and currency units before calculating. Do not claim exact immunization from equal duration alone. A default-free zero maturing on the liability date can match this single nominal cash flow if purchased in the required amount, but it still does not automatically hedge an inflation-linked or differently denominated obligation.
Formula and assumptions
PV=L/(1+y)^T; annual zero: Dmac=T, Dmod=T/(1+y); ΔPV≈−Dmod×PV×Δy
Compare with the model response after drafting
- PV = 100/1.04^5 = 82.1927 m. Liability modified duration = 5/1.04 = 4.807692 years. The candidate approximately matches present value and modified duration; compare exact cash-flow matching before using the index with modified duration 7.
- A +100 bp parallel yield move implies a first-order liability change of −4.807692×82.192711×0.01 = −3.951573 m, before convexity. Matching money duration offsets only the first-order parallel effect; the exact liability PV at 5% is 78.352617 m.
Errors to catch
- Calling a duration match a guarantee against all yield-curve movements.
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