Immunization with Interest Rate Swaps

How to hedge a portfolio's duration and convexity gaps using plain-vanilla interest rate swaps, as implemented in the ALM toolkit.

The Problem

An insurer holds assets (bonds, mortgages, private credit) backing liabilities (annuities, life insurance). If interest rates change, assets and liabilities reprice by different amounts because they have different duration and convexity profiles. The surplus (Assets − Liabilities) swings unpredictably.

Goal: Make surplus insensitive to parallel shifts in the yield curve by closing both the duration gap and the convexity gap.

Interest Rate Swaps as Hedging Instruments

A plain-vanilla fixed-for-floating swap exchanges:

Fixed leg: periodic payments at a pre-agreed rate
Floating leg: periodic payments that reset to market rates

Position	Duration effect	Economic equivalent
Receive-fixed (pay floating)	Adds duration	Long a fixed-rate bond, short a floater
Pay-fixed (receive floating)	Removes duration	Short a fixed-rate bond, long a floater

Swaps are off-balance-sheet — they adjust risk without buying or selling bonds.

Code: InterestRateSwap in src/alm/core.py

Step 1 — Measure the Gaps

Dollar Duration Gap (DV01)

DV01 measures the dollar change in value for a 1 bp parallel rate shift:

\[ \text{DV01} = \frac{PV(y - 0.0001) - PV(y + 0.0001)}{2} \]

The gap to close is:

\[ \Delta\text{DV01} = \text{DV01}_{\text{liabilities}} - \text{DV01}_{\text{assets}} \]

A positive gap means liabilities are more rate-sensitive than assets — rates falling would widen the deficit.

Code: dv01() in src/alm/core.py

Dollar Convexity Gap

Dollar convexity measures how DV01 itself changes as rates move (the second derivative of PV with respect to yield):

\[ C_\$ = \frac{PV(y + h) + PV(y - h) - 2 \cdot PV(y)}{h^2} \]

The gap:

\[ \Delta C_\$ = C_{\$,\text{liabilities}} - C_{\$,\text{assets}} \]

Duration-only hedging leaves the portfolio exposed to large rate moves. Closing the convexity gap fixes this.

Code: dollar_convexity() in src/alm/core.py

Step 2 — Choose Two Hedging Instruments

You need two instruments with linearly independent sensitivity profiles to target both DV01 and convexity simultaneously. A short- tenor and a long-tenor swap work well because:

Both have non-zero DV01 (from their fixed legs)
The longer swap has proportionally more convexity
The ratio of DV01-to-convexity differs between them

Instrument	Why it works
5-year swap	Lower duration, lower convexity per unit notional
10-year swap	Higher duration, higher convexity per unit notional

Important — parallel shift convention for swaps: When computing swap sensitivities, bump the floating rates and the discount rate together. This reflects a true parallel curve shift. If you only bump the discount rate while holding floating rates fixed, an at-par swap (where fixed rate = floating rate = discount rate) will show zero PV change and zero convexity, making the hedge matrix singular.

# Correct: floating rates move with the discount rate
dc_5y = dollar_convexity(
    lambda r: swap_5y.present_value([r] * swap_5y.n_periods, r),
    discount_rate,
) / swap_5y.notional

# Wrong: floating rates are frozen — convexity ≈ 0
dc_5y = dollar_convexity(
    lambda r: swap_5y.present_value(flat_floats, r),
    discount_rate,
) / swap_5y.notional

Step 3 — Solve the 2×2 System

Given per-unit sensitivities of each hedging instrument, solve for the notionals $n_1$ and $n_2$:

\[ \begin{bmatrix} \text{DV01}_1 & \text{DV01}_2 \\ C_{\$,1} & C_{\$,2} \end{bmatrix} \begin{bmatrix} n_1 \\ n_2 \end{bmatrix} = \begin{bmatrix} \Delta\text{DV01} \\ \Delta C_\$ \end{bmatrix} \]

The solution via Cramer's rule:

\[ n_1 = \frac{\Delta\text{DV01} \cdot C_{\$,2} - \Delta C_\$ \cdot \text{DV01}_2} {\text{DV01}_1 \cdot C_{\$,2} - \text{DV01}_2 \cdot C_{\$,1}} \]

\[ n_2 = \frac{\text{DV01}_1 \cdot \Delta C_\$ - C_{\$,1} \cdot \Delta\text{DV01}} {\text{DV01}_1 \cdot C_{\$,2} - \text{DV01}_2 \cdot C_{\$,1}} \]

The denominator is the determinant of the sensitivity matrix. If it is zero (or near-zero), the two instruments are linearly dependent and cannot independently target both gaps — pick instruments with more separation in tenor.

Interpreting the sign of $n$:

Sign	Meaning
$n > 0$	Enter a receive-fixed position of that notional
$n < 0$	Enter a pay-fixed position of $

Code: immunize() in src/alm/core.py

Step 4 — Verify Under Stress

After computing the hedge, re-run the portfolio under parallel rate shocks (e.g. ±100 bp, ±200 bp) and compare surplus with and without the hedge. A well-immunized portfolio should show:

Surplus nearly flat across small shocks (duration matched)
Surplus slightly gaining under large shocks in either direction (positive net convexity)

Worked Example

from alm.core import (
    InterestRateSwap, dv01, dollar_convexity, immunize,
)

DISCOUNT_RATE = 0.04

# 1. Measure gaps (computed from your asset/liability portfolio)
asset_dv01 = sum(dv01(a.present_value, DISCOUNT_RATE) for a in assets)
liab_dv01  = sum(dv01(l.present_value, DISCOUNT_RATE) for l in liabilities)
dd_gap = liab_dv01 - asset_dv01

asset_dc = sum(dollar_convexity(a.present_value, DISCOUNT_RATE) for a in assets)
liab_dc  = sum(dollar_convexity(l.present_value, DISCOUNT_RATE) for l in liabilities)
dc_gap = liab_dc - asset_dc

# 2. Define hedging instruments
swap_5y  = InterestRateSwap(notional=1, fixed_rate=0.04, tenor=5,  pay_fixed=False)
swap_10y = InterestRateSwap(notional=1, fixed_rate=0.04, tenor=10, pay_fixed=False)

# 3. Compute per-unit sensitivities (parallel shift convention)
dd_5y  = swap_5y.dv01([0.04] * swap_5y.n_periods, DISCOUNT_RATE)
dc_5y  = dollar_convexity(
    lambda r: swap_5y.present_value([r] * swap_5y.n_periods, r),
    DISCOUNT_RATE,
)
dd_10y = swap_10y.dv01([0.04] * swap_10y.n_periods, DISCOUNT_RATE)
dc_10y = dollar_convexity(
    lambda r: swap_10y.present_value([r] * swap_10y.n_periods, r),
    DISCOUNT_RATE,
)

# 4. Solve
n1, n2 = immunize(dd_gap, dc_gap, dd_5y, dc_5y, dd_10y, dc_10y)
# n1 = notional for the 5Y swap
# n2 = notional for the 10Y swap

Common Pitfalls

Pitfall	Consequence	Fix
Only matching DV01 (ignoring convexity)	Surplus exposed to large rate moves	Use two instruments and match both
Freezing floating rates when computing swap convexity	Sensitivity matrix is singular (`ValueError`)	Bump floating rates with the discount rate
Using two swaps with very similar tenors	Near-singular matrix, unstable notionals	Separate tenors by at least 3–5 years
Forgetting to re-hedge after asset/liability changes	Gaps drift over time	Re-run periodically or after material changes

Quick Reference

Function	Module	Purpose
`dv01(pv_func, rate)`	`core`	Dollar duration via central difference
`dollar_convexity(pv_func, rate)`	`core`	Dollar convexity via central difference
`immunize(dd_gap, dc_gap, ...)`	`core`	Solve 2×2 hedge for notionals
`InterestRateSwap`	`core`	Swap cashflows, PV, DV01, convexity

Sign	Meaning
\(n > 0\)	Enter a receive-fixed position of that notional
\(n < 0\)	Enter a pay-fixed position of $