Redington Immunization

Define surplus $S(i) = P_A(i) - P_L(i)$, the value of assets minus liabilities. Immunization arranges the assets so that $S$ sits at a local minimum of zero at today's rate — any small rate move, up or down, leaves surplus at zero or better.

By the end you'll be able to state the three Redington conditions, explain why each one zeroes a term of $S$'s Taylor expansion, and predict when a mismatch turns a safe rate move into a loss.

Predict: with assets and liabilities matched in PV and duration, does a small rate move up OR down leave surplus non-negative? Nudge the rate slider either way and check.

A liability of 1000 is due at t = 4 at i₀ = 6%. It's immunized with a 2-year and a 6-year zero-coupon bond. Drag funding ratio to break the PV match, drag duration tilt to shift weight toward one bond and break the duration match, and watch the convexity readout — then nudge the rate to see whether surplus survives.

Funding ratio = 100%

Duration tilt = 0%

Rate nudge Δi = +0.00%

PV match ✓ Duration match ✓ Convexity C_A ≥ C_L ✓

At $i_0$: $S(i_0)$ = $0.00, $D_A$ = 4.00 yrs ($D_L$ = 4.00), $C_A$ = 21.36 ($C_L$ = 17.80). At nudged rate: $S(i_0+\Delta i)$ = $0.00.

S(i) = P_A(i) − P_L(i) — surplus as the rate moves away from i₀

S(i) i₀ = 6%

An insurer or pension fund holds assets to pay future liabilities; both move with interest rates. Redington's insight is to make surplus a local minimum of zero at today's rate, so a small rate move in either direction can't push it negative.

Intuitive

Picture assets and liabilities as two bowls — convex curves — touching at $i_0$ with equal height and equal slope. If the asset bowl is more curved than the liability bowl, it sits above the liability bowl on both sides of the touch point, so assets stay ahead of liabilities for any nearby rate. Matching height and slope alone (PV and duration) only makes the bowls tangent — it doesn't say which one is on top. The curvature comparison is what pins the assets above.

Formal

Let both streams be valued at today's rate $i_0$. Redington immunization requires:

1. PV match: $P_A = P_L$, so $S(i_0) = 0$.
2. Duration match: asset and liability durations are equal (with equal PVs, matching D_mod is the same as matching Macaulay duration; see the portfolio-duration lesson), so $S'(i_0) = 0$.
3. Convexity: $C_A \ge C_L$, so $S''(i_0) \ge 0$.

Together these give $S(i_0)=0$, $S'(i_0)=0$, $S''(i_0)\ge 0$ — the first- and second-derivative test for a local minimum. The Taylor expansion around $i_0$ then gives, for a small $\Delta i$ in either direction,

$S(i_0+\Delta i) \approx \tfrac12 S''(i_0)(\Delta i)^2 \ge 0$.

Convexity is what makes the asset side recover faster than the liability side after a small perturbation — that's the third condition earning its keep.

Applied

A pension fund owes a single benefit payment years from now. Rather than guess which way rates will move, it buys a bracket of bonds — some shorter, some longer than the liability date — chosen so the fund's PV, duration, and convexity all line up with the three conditions above. Whichever way the rate drifts a little, the fund's asset side stays at least as well-funded as its liability side, without anyone having to forecast rates correctly.

Worked example

Liability: 1000 due at $t = 4$, $i_0 = 6\%$. $P_L = 1000 \cdot 1.06^{-4} = 792.09$, $D_L = 4$. Immunize with a 2-year and a 6-year zero-coupon bond, PVs $x$ and $y$:

PV match: $x + y = 792.09$. Duration match: $2x + 6y = 4(792.09)$.

Solving both equations gives $x = y = 396.05$. Two zeros bracketing $t=4$ with equal PV give duration exactly 4 and, being more spread out than the single liability payment, strictly greater convexity — condition 3 holds automatically here. Set funding ratio = 100% and duration tilt = 0% above and check the surplus curve sits flat at zero, bowl-shaped, at $i_0$.

Your turn

Same setup, different numbers: a liability of 1000 due at $t = 6$, $i_0 = 5\%$. $P_L = 1000 \cdot 1.05^{-6} = 746.22$. Immunize with a 3-year and a 9-year zero-coupon bond, PVs $x$ and $y$. PV match: $x + y = 746.22$. Duration match: $3x + 9y = 6(746.22)$, i.e. $x + 3y = 2(746.22)$. Subtracting the PV-match equation: $2y = 746.22$, so $y = $ ____, and $x = 746.22 - y = $ ____.

Reveal the answer

$y = 746.22/2 = 373.11$, and $x = 746.22 - 373.11 = $ 373.11. The bracket is symmetric about $t=6$ (3 and 9 years each 3 years away), so an equal PV split lands the duration exactly on 6, matching $D_L$.

More info — why spreading assets out tends to add convexity

For a PV-weighted set of times $t$ with mean $D$ (the duration), the weighted average of $t(t+1)$ splits into $D(D+1) + \text{Var}(t)$, where $\text{Var}(t)$ is the PV-weighted spread of the payment times around $D$. A single liability payment has $\text{Var}(t) = 0$ — the smallest possible value — so once an asset portfolio matches that same duration, spreading its cash flows out in time (as the worked example's 2-year/6-year bracket does) can only add convexity relative to the single-payment case, never subtract it. That's why condition 3 held "automatically" in the worked example above. See the Marcel Finan guide in Dive deeper for the full derivation and cases where a liability itself is spread out enough to flip the comparison.

Check your understanding

Question 1 of 4

A liability of 1000 is due at $t = 5$ years, valued at $i_0 = 8\%$. You immunize it with a 2-year and a 10-year zero-coupon bond. What fraction of the total PV should go to the 10-year bond to satisfy the duration-match condition?

Question 2 of 4

Assets and liabilities have equal PV and equal duration at $i_0$, but asset convexity is LESS than liability convexity. What happens to surplus for a small rate change in either direction?

Question 3 of 4

Assets have modified duration $D_{mod} = 5.50$ and convexity $C_A = 42$; the liability has the same modified duration $D_{mod} = 5.50$ and convexity $C_L = 38$. Both have equal PV at $i_0$. For a small $\Delta i$, is surplus protected, and why?

Question 4 of 4

Redington immunization is set up correctly (all three conditions hold) at $i_0 = 6\%$. Six months later, without rebalancing, is the portfolio still immunized?

Recap

Surplus $S(i) = P_A(i) - P_L(i)$; immunization makes $i_0$ a local minimum of $S$ at zero.
Three Redington conditions at $i_0$: PV match ($S(i_0)=0$), duration match ($S'(i_0)=0$), and $C_A \ge C_L$ ($S''(i_0)\ge 0$).
Together they give $S(i_0+\Delta i) \approx \tfrac12 S''(i_0)(\Delta i)^2 \ge 0$ for a small rate move in either direction.
Protection is local and temporary: it covers only small, parallel rate shifts at a single point in time, and drifts as time passes and yields move — the match must be re-established periodically.

Dive deeper

Bond Immunization — SOA Exam FM Study Notes (AnalystPrep) State and apply the three Redington immunization conditions.
Marcel Finan — Exam FM study guide (PDF) Work immunization problems with detailed solutions.

Sources