Bond Risk Measures and Immunization › Liability-Driven Investing (LDI)
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Liability-Driven Investing extends immunization concepts to manage portfolios against complex liability streams. Rather than targeting a single future payment, LDI addresses situations where institutions must meet a series of obligations over time. The framework shifts the investment objective from maximizing returns to ensuring that assets will be sufficient to meet liabilities under various market conditions.
Pension funds provide a classic example. A defined-benefit pension fund must pay retirees monthly benefits potentially spanning decades. The present value of these liabilities depends on interest rates. When rates fall, the present value of fixed liability payments rises. Funding can weaken if liabilities rise faster than assets in percentage terms; whether a deficit develops also depends on the initial funding position and the size of the rate move. A fund that appeared adequately funded at high interest rates may suddenly face a deficit when rates decline, even if the actual benefit obligations have not changed.
Duration Gap
The duration gap measures the mismatch between asset duration and liability duration weighted by the inverse funding ratio:
where:
- : duration of the asset portfolio
- : duration of the liabilities
- : present value of assets
- : present value of liabilities
- : the inverse of the funding ratio
The duration gap measures first-order exposure of dollar surplus . Under a common small decimal yield change, . A positive gap means rising yields reduce dollar surplus; a negative gap means rising yields increase it. This concerns dollar surplus, not the percentage changes in asset and liability values.
A zero gap equates asset and liability money durations, so dollar surplus has zero first-order yield sensitivity. The funding ratio has a different derivative: . Its first-order sensitivity vanishes when the two modified durations are equal. For an overfunded fund, that is generally not the same condition as a zero surplus duration gap.
# Example: Pension fund LDI analysis
asset_value = 100_000_000 # $100 million in assets
liability_value = 95_000_000 # $95 million in liabilities (overfunded)
asset_duration = 6.5
liability_duration = 12.0
funding_ratio = asset_value / liability_value
duration_gap = asset_duration - (liability_value / asset_value) * liability_duration
print("Pension Fund LDI Analysis")
print("=" * 50)
print(f"Asset Value: ${asset_value:,.0f}")
print(f"Liability Value: ${liability_value:,.0f}")
print(f"Funding Ratio: {funding_ratio:.2%}")
print(f"Asset Duration: {asset_duration:.1f} years")
print(f"Liability Duration: {liability_duration:.1f} years")
print(f"Duration Gap: {duration_gap:.2f}")
print()
# Impact of 1% rate change
rate_change = 0.01
asset_change = -asset_duration * asset_value * rate_change
liability_change = -liability_duration * liability_value * rate_change
surplus_change = asset_change - liability_change
print("Impact of +1% Rate Change:")
print(f" Asset Value Change: ${asset_change:,.0f}")
print(f" Liability Value Change: ${liability_change:,.0f}")
print(f" Surplus Change: ${surplus_change:,.0f}")
print()
print(
f"New Funding Ratio: {(asset_value + asset_change) / (liability_value + liability_change):.2%}"
)Pension Fund LDI Analysis ================================================== Asset Value: $100,000,000 Liability Value: $95,000,000 Funding Ratio: 105.26% Asset Duration: 6.5 years Liability Duration: 12.0 years Duration Gap: -4.90 Impact of +1% Rate Change: Asset Value Change: $-6,500,000 Liability Value Change: $-11,400,000 Surplus Change: $4,900,000 New Funding Ratio: 111.84%
The negative duration gap (-4.9) indicates that liabilities are more rate-sensitive than assets. A 1% rate increase benefits the fund because liabilities fall more than assets, improving the funding ratio. Conversely, falling rates would damage the fund's position. This analysis highlights the interest rate risk that pension funds face when their asset duration does not match their liability duration.
LDI Strategies
Pension funds and insurers employ several LDI approaches to manage the duration gap and protect their funding status:
-
Cash flow matching: Structure bond portfolios so that coupon and principal payments exactly match liability payment dates in both timing and amount. This approach eliminates reinvestment risk entirely because each cash inflow is immediately used to fund an outflow. However, cash flow matching may be expensive or impossible to implement for very long-dated liabilities, as appropriate bonds may not exist or may trade at premium prices.
-
Duration matching: Match portfolio duration to liability duration, accepting some reinvestment risk in exchange for flexibility. This approach allows the use of a broader range of bonds and can be implemented at lower cost than perfect cash flow matching. The tradeoff is that the fund remains exposed to non-parallel yield curve shifts.
-
Key rate duration matching: Match sensitivity to specific points on the yield curve, and protects against non-parallel curve movements. Instead of a single duration number, key rate durations measure sensitivity to changes at specific maturities (such as the 2-year, 5-year, 10-year, and 30-year points). Matching key rate durations across assets and liabilities gives more stable protection but requires more sophisticated analysis and implementation.
-
Liability-driven benchmarking: Compare the portfolio's cash-flow-adjusted percentage return with the percentage change in a consistently valued liability benchmark. Asset returns above that benchmark indicate relative outperformance, even when both percentage changes are negative. Contributions and benefit payments must be handled consistently, and liability remeasurement must be distinguished from investment performance. This comparison is one performance measure, not a complete judgment of a plan's success.
def ldi_scenario_analysis(
asset_value,
liability_value,
asset_duration,
liability_duration,
yield_scenarios,
):
"""Analyze funding ratio under different yield scenarios."""
results = []
for dy in yield_scenarios:
new_assets = asset_value * (1 - asset_duration * dy)
new_liabilities = liability_value * (1 - liability_duration * dy)
new_surplus = new_assets - new_liabilities
new_funding_ratio = new_assets / new_liabilities
results.append(
{
"yield_change": dy,
"assets": new_assets,
"liabilities": new_liabilities,
"surplus": new_surplus,
"funding_ratio": new_funding_ratio,
}
)
return results
The current portfolio's funding ratio improves when rates rise because its liability percentage sensitivity exceeds its asset sensitivity. The surplus-matched portfolio holds dollar surplus constant in this linear scenario model, but its funding ratio still changes. The added equal-duration series instead holds the funding ratio constant within the model. Neither construction removes convexity, credit, liquidity or general curve-shape risk, and the finite-shock curves use approximate rather than exactly repriced values.