Re-Immunization After the Valuation Rate Changes

Why a Rate Change Requires a Fresh Redington Test

Redington immunization is a local condition at a specified valuation rate. If that base rate changes, each cash flow receives a different present-value weight. As a result, the asset and liability present values, durations, and convexities must all be recalculated at the new rate. A portfolio that satisfied the conditions at its original rate does not automatically satisfy them at the new one.

This distinction matters even when the change in rate is small. Present values may remain close, and the asset convexity may still exceed the liability convexity, while the present-value or duration condition has ceased to hold. For a portfolio to qualify at the new base rate, all required conditions must be checked there.

Definition: For dated cash flows $C_t$ at times $t=1,\ldots,n$ and an effective annual valuation rate $i>-1$, the present value is $P(i)=\sum_{t=1}^{n}C_t(1+i)^{-t}$. For positive cash flows, the Macaulay duration is $D(i)=\sum_{t=1}^{n}tC_t(1+i)^{-t}/P(i)$, and the convexity is $\Gamma(i)=\sum_{t=1}^{n}t(t+1)C_t(1+i)^{-t}/[P(i)(1+i)^2]$. These definitions apply separately to assets and liabilities.

For a positive-cash-flow asset portfolio and a positive-cash-flow liability stream, the Redington conditions at a base rate $i$ are: equal present values, equal Macaulay durations, and strictly greater asset convexity than liability convexity. The first two conditions make the surplus locally level and give it zero first derivative; the strict convexity condition makes it a local minimum. If any of the first two conditions fails, a convexity advantage alone does not establish Redington immunization.

Use the same effective annual rate and timing convention for both sides. In particular, do not discount one side using the old base rate while evaluating the other at the new rate. The formulas here assume dated, deterministic cash flows and a single effective annual yield applied consistently to every payment date.

Why the Measures Must Be Recomputed

The rate enters each present-value term as a power of $1+i$. Therefore, a rate change alters both the total present value and the relative weights assigned to early and late cash flows. Duration and convexity are weighted measures: they depend on those present-value weights, not just on the payment dates.

\frac{dP}{di}=-\frac{P(i)D(i)}{1+i}, \qquad \frac{d^2P}{di^2}=P(i)\Gamma(i).

These identities follow by differentiating the cash-flow present-value sum term by term. For one term, $C_t(1+i)^{-t}$, the first derivative is $-tC_t(1+i)^{-t-1}$. Summing these derivatives gives $-\sum tC_t(1+i)^{-t-1}$, which equals $-P(i)D(i)/(1+i)$ by the definition of $D(i)$. Differentiating a second time gives $\sum t(t+1)C_t(1+i)^{-t-2}$, equal to $P(i)\Gamma(i)$ by the definition of $\Gamma(i)$. Thus the measures describing local rate sensitivity are themselves functions of the valuation rate.

The practical consequence is straightforward: after changing the base rate, replace the old rate with the new one in every present-value term, and then recompute the measures from those new terms. Do not merely compare the old duration or convexity figures with one another.

Worked Examples

Worked Example: A Previously Matched Portfolio at a New Yield

At an old effective annual yield of 5%, assets pay $52.50 at time 1 and $57.88125 at time 3. A liability pays $110.25 at time 2. The portfolio satisfies Redington conditions at the old yield. Revalue it at a new effective annual yield of 4%, and determine whether it still qualifies.

Method. Recalculate each present value at 4%, then calculate duration and convexity from the new present-value contributions. The common mistake is to assume that an old match remains a match after the rate changes.

At 5%, the asset present-value contributions are $52.50/1.05=50$ and $57.88125/(1.05)^3=50$. The asset present value is $100, its duration is $(1(50)+3(50))/100=2$, and its convexity is $700/[100(1.05)^2]=6.349206$. The liability present value is $110.25/(1.05)^2=100$, with duration 2 and convexity $6/(1.05)^2=5.442177$. Thus the three conditions hold at 5%.

At 4%, the asset contributions are:

PV_{A,1}=\frac{52.50}{1.04}=50.480769, \qquad PV_{A,3}=\frac{57.88125}{(1.04)^3}=51.456220.

The asset present value is $50.480769+51.456220=101.936989$. The liability present value is $110.25/(1.04)^2=101.932323$. Therefore, the present values differ by approximately $0.004666, so the present-value condition fails.

The asset duration is:

D_A(0.04)= \frac{1(50.480769)+3(51.456220)}{101.936989} =\frac{204.849429}{101.936989} \approx 2.0096.

The liability duration remains 2, because its only payment is at time 2. The duration condition also fails. The asset convexity is:

\Gamma_A(0.04)= \frac{2(50.480769)+12(51.456220)} {101.936989(1.04)^2} = \frac{718.436184}{110.255?} \approx 6.517.

Here the denominator before rounding is $101.936989(1.0816)$, approximately $110.255?$. The liability convexity is $6/(1.04)^2\approx5.547337$. Thus asset convexity is greater, but the portfolio does not qualify at 4% because present value and duration are not matched. Convexity cannot repair either failed condition.

Worked Example: Diagnosing a Mismatch at the New Base Rate

At a new effective annual yield of 6%, assets pay $60 at times 1 and 3, and a liability pays $120 at time 2. Test the present-value condition and compare the durations.

Method. Discount every payment at 6%. This example also guards against an arithmetic trap: the third-year asset payment is discounted by $(1.06)^3$, not by $(1.06)^2$.

The asset present value is:

PV_A=\frac{60}{1.06}+\frac{60}{(1.06)^3} =56.603774+50.377156 =106.980930.

The liability present value is:

PV_L=\frac{120}{(1.06)^2} =\frac{120}{1.1236} =106.799573.

The present values are unequal, so the portfolio fails the first Redington condition. For completeness, the asset duration is:

D_A= \frac{1(56.603774)+3(50.377156)}{106.980930} = \frac{207.735242}{106.980930} \approx1.9418.

The liability duration is 2, since it has one payment at time 2. Duration also fails to match. Even if a separate convexity calculation showed greater asset convexity, the portfolio would not qualify at this base yield. The correction to the common discounting error is to use asset present value $106.980930 and liability present value $106.799573; neither value should be substituted for the other.

Worked Example: Re-Immunizing a Single Liability at a New Yield

A liability pays $120 at time 2. At a new effective annual valuation yield of 6%, construct two positive asset payments at times 1 and 3 that satisfy the Redington conditions. Use equal asset present-value contributions to match the liability duration.

Method. First find the liability present value. Split that amount equally between the two asset dates, so the asset duration is the midpoint, time 2. Then check convexity. The mistake to avoid is splitting the nominal asset payments equally; the split must be made in present-value terms.

The liability present value is $120/(1.06)^2=106.799573$. Half of this is $53.399786$ to the nearest dollar-micro. To produce that present value at time 1, the asset payment must be $53.399786(1.06)=56.603774$. At time 3, it must be $53.399786(1.06)^3=63.60$. The latter is exactly $63.60 using the unrounded half-present-value amount.

Discounting the asset payments gives present-value contributions of $56.603774/1.06=53.399786$ and $63.60/(1.06)^3=53.399786$, subject only to displayed rounding. Their sum equals the liability present value, and the asset duration is:

D_A=\frac{1(53.399786)+3(53.399786)} {2(53.399786)}=2=D_L.

The asset convexity is $14(53.399786)/[2(53.399786)(1.06)^2]=7/(1.1236)\approx6.229975$. The liability convexity is $6/(1.06)^2\approx5.339623$. Since $6.229975>5.339623$, the present-value, duration, and strict convexity conditions all hold at 6%. Rounding the displayed asset payment at time 1 to six decimals does not change the conclusion; the construction uses the unrounded values.

A Reliable Re-Immunization Procedure

When the base valuation rate changes, organize the work so that no old-rate measure is carried into the new test. Use the following sequence for positive dated cash flows:

Fix the new base rate.
State the effective annual yield and ensure the same rate and timing convention are used for assets and liabilities.

Recalculate each present value.
Discount every payment at the new rate and total the asset and liability present values separately.

Recalculate duration and convexity.
Use the new present-value contributions as weights. Do not reuse values computed at the old base rate.

Apply all three conditions.
Check equal present values, equal durations, and strictly greater asset convexity. A failure of either equality means the portfolio is not Redington-immunized at this rate.

Reconstruct if needed.
If the conditions fail, adjust asset amounts or weights and repeat the full test at the new base rate.

A small change in yield can produce a small present-value mismatch, as the examples illustrate. Small does not mean zero: the equality conditions are mathematical conditions, not approximate labels. In a practical calculation, retain unrounded values through the comparisons and round only when presenting results. This topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Common Mistakes

Keeping old duration or convexity values. Both depend on present-value weights, which change with the base yield.
Checking convexity first and stopping there. Greater asset convexity is only the third condition; it does not compensate for unequal present values or durations.
Using inconsistent yield conventions. A rate must be interpreted consistently as an effective annual yield for all dated cash flows in these formulas.
Discounting by the wrong number of periods. A payment at time 3 is divided by $(1+i)^3$, even when another payment occurs at time 2.
Rounding before testing equality. Retain full precision in present values and weighted sums; display rounded results only after the checks.

Key takeaway: Redington immunization is rate-specific. At a changed valuation rate, independently recalculate asset and liability present values, durations, and convexities. The portfolio qualifies only if the two present values and durations match and asset convexity is strictly greater.

Check Your Understanding

For each problem, use the stated effective annual yield and show enough intermediate work to verify the Redington test.

At 5%, assets pay $105 at time 1 and $115.7625 at time 3; a liability pays $220.50 at time 2. Find the asset and liability present values, durations, and convexities. Does the portfolio satisfy Redington conditions? (Answer: Both present values are $200; both durations are 2; asset convexity is 6.3492 and liability convexity is 5.4422, so it qualifies.)
At 5%, assets pay $40 at time 1 and $80 at time 3, while a liability pays $120 at time 2. Does the present-value condition hold? (Answer: No. $PV_A=40/1.05+80/(1.05)^3\approx107.202$, while $PV_L=120/(1.05)^2\approx108.844$.)
At a 0% effective annual yield, assets pay $100 at times 1 and 3, and a liability pays $200 at time 2. Check the three Redington conditions. (Answer: Both present values are $200 and both durations are 2; asset convexity is 7 and liability convexity is 6, so it qualifies.)
A liability pays $300 at time 2, and the new effective annual yield is 5%. Construct asset payments at times 1 and 3 using equal present-value contributions. (Answer: The liability present value is $300/(1.05)^2$; the asset payments are approximately $142.86 at time 1 and $157.50 at time 3. The present values and durations match, and asset convexity exceeds liability convexity.)

What You'll Learn