Nonparallel Yield-Curve Changes and Redington Limits

Why a Parallel-Rate Match Has Limits

Redington immunization is a local result: under its standard conditions, matching present value and first-order sensitivity, with asset convexity exceeding liability convexity, protects surplus against sufficiently small parallel changes in the valuation yield. A parallel change moves the yields used for every maturity by the same amount. A real yield curve can instead change by different amounts at different maturities. The parallel-shift conditions do not, by themselves, control that pattern of changes.

To see the distinction, it helps to track each dated cash flow separately. A portfolio can have zero present-value surplus and zero sensitivity to a common yield movement, while still having a nonzero sensitivity to a movement concentrated at one maturity. The remaining surplus exposure is maturity-specific.

Model and assumptions: Cash flows are deterministic, dated at integer times $t=1,2,\ldots,n$, and have no embedded options or default risk. Each cash flow at time $t$ is discounted using an annual effective spot rate $i_t$, so its present value is divided by $(1+i_t)^t$. For a parallel shift, all the $i_t$ change by the same amount. This spot-rate setup is a diagnostic model; it does not assume that arbitrary maturity-specific shocks describe every possible market yield-curve movement.

Let $A_t$ be the asset cash flow and $L_t$ the liability cash flow at time $t$, and define the net cash flow $N_t=A_t-L_t$. At a given spot curve, the surplus present value is the present value of the net cash flows. A positive surplus means assets have greater present value than liabilities.

S=\sum_{t=1}^{n}\frac{N_t}{(1+i_t)^t}

If the curve is flat at an annual effective yield $i$, this reduces to the usual single-yield valuation. The standard local Redington conditions are: $S=0$; the derivative of surplus with respect to a common change in $i$ is zero; and the second derivative of surplus with respect to that common yield is positive. These conditions concern a common yield change, not a separate change at each maturity.

From Parallel Sensitivity to Maturity-Specific Sensitivity

Differentiate the present value of one net cash flow $N_t(1+i_t)^{-t}$ with respect to its own spot rate. Its rate sensitivity is $-tN_t(1+i_t)^{-t-1}$. Thus, for small changes $\Delta i_t$, the first-order change in surplus is the sum of each cash flow's sensitivity multiplied by its own rate change.

\Delta S\ \approx\ -\sum_{t=1}^{n} \frac{tN_t}{(1+i_t)^{t+1}}\,\Delta i_t

The approximation is first-order: it omits terms involving squared changes and products of changes. Define the signed maturity sensitivity $K_t=tN_t/(1+i_t)^{t+1}$. For a parallel shift, $\Delta i_t=\Delta i$ for every $t$, so the first-order surplus change is $-\Delta i\sum_t K_t$. A first-order parallel match makes this sum zero. For nonparallel movements, the change instead depends on the weighted sum $\sum_t K_t\Delta i_t$. A zero unweighted sum of sensitivities does not force that weighted sum to be zero.

\text{Parallel match: }\sum_t K_t=0 \qquad\not\Rightarrow\qquad \text{Nonparallel match: }\sum_t K_t\Delta i_t=0

This is the central limitation. Parallel matching controls one direction of movement in the vector of maturity-specific rates. It does not generally control other directions. Exact repricing after a shock can differ from the first-order estimate because of curvature; the estimate is a diagnostic for small changes, not a substitute for repricing.

Why Redington Convexity Does Not Remove This Limitation

For a flat annual effective yield $i$, the first and second derivatives of surplus are obtained by differentiating each dated cash flow's present value. The first derivative is $-\sum_t tN_t/(1+i)^{t+1}$; the second is $\sum_t t(t+1)N_t/(1+i)^{t+2}$. Redington's first-order condition sets the first expression to zero, and its strict convexity condition requires the second expression to be positive.

\frac{dS}{di} =-\sum_t\frac{tN_t}{(1+i)^{t+1}}, \qquad \frac{d^2S}{di^2} =\sum_t\frac{t(t+1)N_t}{(1+i)^{t+2}}

These derivatives describe how surplus changes along the single path in which every maturity yield moves together. The positive second derivative supports local protection along that path when the other Redington conditions hold. It does not imply that the portfolio has zero sensitivity to every possible pattern of maturity-specific shocks, nor does it guarantee a favorable outcome for a particular nonparallel movement.

Key takeaway: Redington matching is not the same as matching sensitivity separately at every maturity. Under a nonparallel change, calculate or estimate the surplus using the maturity-specific rate movements; do not infer the result from the parallel-shift conditions alone.

Worked Redington Match and a Single-Maturity Shock

Worked Example: A Matched Portfolio with an Unmatched Maturity Exposure

A hypothetical portfolio owes $84$ at the end of year 2. It holds assets paying $40$ at the end of year 1 and $44.10$ at the end of year 3. The valuation yield is a flat annual effective rate of $5\%$. Verify the present-value and first-order parallel matches, then find the exact surplus change if only the year-1 spot rate rises by 1 percentage point, from $5\%$ to $6\%$.

Method. First value each cash flow at the base yield, then compare its first-order parallel sensitivity. For the shock, reprice the affected year-1 asset cash flow exactly. This avoids the mistake of treating a maturity-specific change as a parallel yield shift.

Base present values. The discount factor is $v=1/1.05=20/21$. The year-1 asset has present value $40v=38.095238$. The year-3 asset has present value $44.10v^3=38.095238$. The liability has present value $84v^2=76.190476$. Therefore, asset present value is $38.095238+38.095238=76.190476$, equal to liability present value.

Parallel first-order check. The signed sensitivity magnitudes for the two asset cash flows are $1(40)/(1.05)^2=36.281179$ and $3(44.10)/(1.05)^4=108.843537$. The liability's corresponding magnitude is $2(84)/(1.05)^3=145.124716$. Assets and liabilities have equal total sensitivity because $36.281179+108.843537=145.124716$. Thus the first derivative of surplus for a parallel change is zero.

Exact maturity-specific repricing. Only the year-1 asset changes in value. Its new present value is $40/1.06=37.735849$, a change of $37.735849-38.095238=-0.359389$. The year-3 asset and year-2 liability are unchanged. The initial surplus was zero, so the new surplus is $-0.359389$, or -$0.36 to the nearest cent.

Check. A rise in the year-1 spot rate reduces the value of the year-1 asset, while leaving the other cash flows unchanged; the negative surplus change has the expected sign. The portfolio is matched for a parallel first-order movement but not for this isolated year-1 shock.

First-Order Cancellation Can Still Leave a Residual

A nonparallel shock can also be chosen so that its first-order effects cancel. That cancellation is not an exact immunity result: the omitted second-order terms can still change surplus. The next example isolates that distinction.

Worked Example: Unequal Shocks with a First-Order Offset

Use the same hypothetical assets, liability, and $5\%$ base annual effective yield. The year-1 spot rate rises by 1 percentage point to $6\%$, the year-2 spot rate is unchanged, and the year-3 spot rate falls by one-third of a percentage point to $4\frac{2}{3}\%$. Calculate the exact change in surplus.

Method. Check first-order cancellation using the asset sensitivities, then reprice both assets exactly. This avoids assuming that a zero first-order estimate means the exact surplus change is zero.

The first-order asset sensitivity magnitudes from the preceding example are $36.281179$ at year 1 and $108.843537$ at year 3. Their weighted rate changes are $36.281179(0.01)=0.362812$ and $108.843537(-0.003333333)=-0.362812$, to the shown precision. The year-2 liability does not change rate. The first-order change therefore cancels.

Now reprice exactly. The year-1 asset changes by $40/1.06-40/1.05=-0.359389$. The year-3 asset's new present value is $44.10/(1.046666667)^3$. Relative to its base present value $38.095238$, this is multiplied by $(1.05/1.046666667)^3=(315/314)^3$, so the year-3 asset's change is $38.095238((315/314)^3-1)$, approximately $+0.365128$.

The liability remains at $76.190476$. The exact surplus change is therefore $-0.359389+0.365128=+0.005739$, or +$0.01 to the nearest cent. The initial surplus was zero, so this is also the final surplus to the stated rounding.

Check. The first-order estimate is zero, but exact repricing gives a small positive residual. This is consistent with the approximation omitting higher-order terms; the sign and size here are specific to these cash flows and shocks, not a general guarantee.

Exact Repricing Across a Changed Curve

For a specified nonparallel curve, direct repricing is straightforward: use the new spot rate for each cash flow's maturity, calculate each present value, and subtract liability value from asset value. This answers the question for the stated curve rather than extrapolating from a local parallel-rate condition.

Worked Example: Repricing after a Tilt in Spot Rates

A hypothetical portfolio has assets of $40$ at year 1 and $44.10$ at year 3, against a liability of $84$ at year 2. The initial curve is flat at an annual effective $5\%$. The new spot rates are $6\%$ at year 1, $5\%$ at year 2, and $4\%$ at year 3. Find the new surplus.

Method. Reprice each dated cash flow at its own new spot rate. The common mistake is to apply one selected yield to all maturities, which would erase the nonparallel feature in the data.

The new year-1 asset value is $40/1.06=37.735849$. The new year-3 asset value is $44.10/(1.04)^3=39.204739$. Total asset value is $37.735849+39.204739=76.940588$.

The year-2 liability is still discounted at $5\%$, so its value is $84/(1.05)^2=76.190476$. The new surplus is $76.940588-76.190476=0.750112$, or $0.75 to the nearest cent.

Check. The year-1 rate increase lowers the short asset's value, but the year-3 rate decrease raises the long asset's value. In this case the latter effect is larger, leaving positive surplus. A different pattern of maturity-specific movements could produce a negative surplus instead.

Exam Trap: Reusing a Parallel-Shock Conclusion

A common error is to verify that present value and duration match, note that assets have greater convexity, and then conclude that surplus is protected from any small yield-curve movement. Those checks establish local Redington protection for the specified parallel-yield model. They do not establish that $\sum_t K_t\Delta i_t=0$ for a nonparallel shock. In particular, the calculation $dS/di=0$ uses one common rate variable; it cannot be substituted for a set of distinct changes $\Delta i_t$.

For a stated curve movement, use the maturity-specific first-order formula as a diagnostic when the changes are small, and use exact repricing when the new rates are given. If the problem asks whether the portfolio is protected under a general set of nonparallel movements, the appropriate conclusion is that a parallel Redington match alone is insufficient. Do not claim immunity without additional maturity-specific matching information.

Identify the rate model.
Determine whether one common yield changes or whether the data specify a different spot-rate change for each maturity.

Check the base match.
Confirm the initial asset and liability present values are equal before interpreting a surplus change.

Use the right sensitivity.
For a small nonparallel move, weight each maturity-specific rate change by that maturity's signed sensitivity.

Reprice when possible.
Discount every cash flow using its new maturity-specific spot rate, then calculate assets minus liabilities.

Check Your Understanding

Assume deterministic annual cash flows with no embedded options or default risk. For numerical answers, round dollar amounts to the nearest cent.

A portfolio has assets of $30$ at year 1 and $33.075$ at year 3, and a liability of $63$ at year 2. The initial flat annual effective yield is $5\%$. If only the year-1 spot rate rises to $6\%$, what is the exact surplus change? (Answer: $-$0.27$)
In a portfolio, the signed maturity sensitivities sum to zero for a parallel rate change. Does this alone imply zero first-order surplus change for every nonparallel rate movement? (Answer: No; the maturity sensitivities must be weighted by the separate rate changes.)
At a $5\%$ base yield, net cash flows are $20$ at year 1, $-42$ at year 2, and $22.05$ at year 3. Rates change by $+0.20$ percentage point at year 1, no change at year 2, and $-0.10$ percentage point at year 3. What is the first-order change in surplus? (Answer: approximately $+$0.02$)
A portfolio has assets of $40$ at year 1 and $44.10$ at year 3, with a liability of $84$ at year 2. Starting from a flat $5\%$ annual effective curve, the new year-1, year-2, and year-3 spot rates are $6\%, 5\%, 4\%$, respectively. What is the exact new surplus? (Answer: $0.75)
Why does a positive second derivative of surplus with respect to a common annual effective yield not prove protection against every nonparallel curve movement? (Answer: It describes curvature only along the path where all maturity yields move together; a nonparallel movement uses separate maturity sensitivities and can leave a nonzero first-order or higher-order surplus change.)

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