Convexity, Key Rates, and Immunization Limits

Why a Duration Match May Not Be Enough

A duration-based hedge is designed to offset the first-order effect of a small, parallel change in yields. That is useful, but it does not guarantee that asset and liability values move together when yields change by a material amount or when different parts of the yield curve move by different amounts. Convexity describes the curvature of price as yields change; key-rate duration separates sensitivity across selected curve maturities. Together, they help identify risks a single duration number conceals.

The discussion below assumes market values are measured at a common valuation date. Unless stated otherwise, price changes from duration and convexity are local approximations, not exact forecasts. A full repricing under the specified curve scenario is preferable when the shift is large or the cash flows are nonlinear.

Convexity and the Price Approximation

Definition: For a bond with deterministic cash flows $CF_t$ at integer times $t$, priced using an annual effective yield $y$, its price is $P(y)=\sum_t CF_t/(1+y)^t$. Its modified duration is $D=-P^{-1}dP/dy$, and its convexity is $C=P^{-1}d^2P/dy^2$. With this convention, convexity is measured in squared years, and the local price-change approximation is $\Delta P/P \approx -D\Delta y+\frac{1}{2}C(\Delta y)^2$.

The yield change $\Delta y$ is in decimal form: a rise of 50 basis points is $0.005$, not $50$. Duration gives the linear component of the change. Convexity adds a second-order component. For an ordinary option-free bond with positive cash flows, price is typically a decreasing, convex function of yield, so $D$ and $C$ are positive. The approximation should not be treated as exact, particularly for a large yield change.

For deterministic cash flows discounted at one annual yield, the derivatives give the standard formulas:

D=\frac{\sum_t t\,CF_t(1+y)^{-t-1}}{P(y)}, \qquad C=\frac{\sum_t t(t+1)\,CF_t(1+y)^{-t-2}}{P(y)}.

These are yield-based measures for the stated cash-flow model. For a portfolio valued from a yield curve, a single-yield formula does not describe every possible curve movement; key-rate sensitivities or direct scenario repricing are needed to examine nonparallel changes.

Worked Example: Improving a Zero-Coupon Price Estimate

A hypothetical two-year zero-coupon bond pays $100 at maturity. Its annual effective yield is 5.00%. Estimate the price change if yield rises by 1.00 percentage point, using duration and convexity, then check against exact repricing. The cash flow is certain, and the yield is compounded annually.

Method. Calculate the initial price, duration, and convexity under the annual-yield model, apply the second-order approximation, and compare it with the new discounted value. This avoids treating a curved price-yield relationship as entirely linear.

Initial price: $P=100/(1.05)^2=100/1.1025=\$90.7029$ before final rounding. For a two-year zero, $D=2/(1.05)=1.9048$ years and $C=2(3)/(1.05)^2=6/1.1025=5.4422$ squared years.

The yield change is $\Delta y=0.0100$. The estimated percentage change is $-1.9048(0.0100)+\frac{1}{2}(5.4422)(0.0100)^2=-0.0190476+0.0002721=-0.0187755$, or $-1.8776\%$. The estimated dollar change is $\$90.7029(-0.0187755)=-\$1.7030$, so the estimated new price is $\$88.9999$.

Exact repricing gives $100/(1.06)^2=\$88.9996$. The exact change is $\$88.9996-\$90.7029=-\$1.7033$. Rounded to the nearest cent, the estimate is $-\$1.70$ and the exact change is $-\$1.70$. The close agreement is a check for this particular small shift, not a guarantee for larger shifts.

Convexity in an Asset-Liability Hedge

For a small parallel shift, let $A$ and $L$ be asset and liability market values, with durations $D_A,D_L$ and convexities $C_A,C_L$. Surplus is $S=A-L$. Applying the local price approximation separately to both values gives:

\Delta S \approx (-A D_A+L D_L)\Delta y +\frac{1}{2}(A C_A-L C_L)(\Delta y)^2.

A duration match satisfies $A D_A=L D_L$, cancelling the first-order term for a small parallel shift. It does not necessarily cancel the second-order term. If asset and liability values are equal and durations match, but asset convexity exceeds liability convexity, this approximation predicts a positive surplus change for either direction of a small parallel shift. This is a local conclusion under the stated model; it is not a promise that surplus will improve under every market scenario.

Worked Example: Testing a Duration-Matched Balance Sheet

A hypothetical insurer has assets worth $100 million and liabilities worth $100 million. Both have modified duration 5.0 years. Their convexities are 40 and 25 squared years, respectively. Estimate the surplus change for a parallel yield rise of 1.00 percentage point. Use the second-order approximation and assume the same annual-yield shift applies to both sides.

Method. Apply the approximation to assets and liabilities separately, then subtract the liability change from the asset change. The common error is to stop after noting that the durations match; matching duration cancels only the first-order term.

For the assets, the percentage change is $-5(0.01)+\frac{1}{2}(40)(0.01)^2=-0.0500+0.0020=-0.0480$, or $-4.80\%$. The estimated asset value change is $-\$4.800$ million.

For the liabilities, the percentage change is $-5(0.01)+\frac{1}{2}(25)(0.01)^2=-0.0500+0.00125=-0.04875$, or $-4.875\%$. The estimated liability value change is $-\$4.875$ million.

Thus $\Delta S=\Delta A-\Delta L=-\$4.800\text{ million}-(-\$4.875\text{ million})=+\$0.075\text{ million}$, or a surplus increase of $75,000. Directly, the duration terms cancel and the convexity term is $\frac{1}{2}(\$100\text{ million})(40-25)(0.01)^2=\$75{,}000$, which checks the result. This is an approximation, not a full repricing.

Key-Rate Duration and Curve Twists

Definition: A key-rate duration measures the local percentage price sensitivity to a change in yield at one selected curve maturity, with the shift shape at other maturities specified as part of the method. For key-rate duration $K_j$ at key rate $j$, the local approximation is $\Delta P/P\approx-\sum_j K_j\Delta y_j$, where each $\Delta y_j$ is the decimal change applied using that key’s defined shift shape.

Key-rate durations are not independent of their scenario construction: the selected maturities and the way a bump is interpolated between them matter. When the key-rate shift shapes form a consistent representation of a parallel shift, their sum approximates the parallel-shift duration. They show how sensitivity is distributed, rather than only reporting its total.

For assets and liabilities with values $A,L$ and key-rate durations $K_{A,j},K_{L,j}$, respectively, the first-order surplus approximation is:

\Delta S \approx -\sum_j (A K_{A,j}-L K_{L,j})\Delta y_j.

A parallel duration match makes the sum of the net key-rate exposures zero, but each exposure at an individual maturity may still be nonzero. Opposing movements at different curve points can therefore change surplus even if the parallel duration calculation indicates a match.

Worked Example: A Parallel Match Under a Curve Twist

A hypothetical portfolio has assets worth $100 million and liabilities worth $100 million. Their key-rate durations, in years, are: assets, 1.0 at two years and 4.0 at ten years; liabilities, 3.0 at two years and 2.0 at ten years. Estimate the first-order surplus change if the two-year yield rises by 50 basis points while the ten-year yield falls by 50 basis points. Assume the reported key-rate durations use shift shapes appropriate to these two scenario moves.

Method. Calculate the net asset-minus-liability dollar exposure at each key rate and apply the corresponding yield change. This avoids incorrectly treating a twist as a single parallel shift.

At two years, the net exposure is $\$100\text{ million}(1.0)-\$100\text{ million}(3.0)=-\$200\text{ million-years}$. At ten years, it is $\$100\text{ million}(4.0)-\$100\text{ million}(2.0)=+\$200\text{ million-years}$. The total net exposure is zero, so a parallel shift would have no first-order effect.

For the stated twist, $\Delta y_2=+0.005$ and $\Delta y_{10}=-0.005$. Therefore $\Delta S\approx- [(-\$200\text{ million-years})(0.005)+(\$200\text{ million-years})(-0.005)]$ $=-[-\$1\text{ million}-\$1\text{ million}]=+\$2\text{ million}$. The result is a first-order estimate of surplus increasing by $2 million. The zero total duration exposure and nonzero twist result are consistent: the two key-rate exposures offset for a parallel move but reinforce under this twist.

What an Immunization Check Can and Cannot Establish

Immunization is conditional on the risks and assumptions represented in the hedge measures. A duration match is a local condition for a parallel move; convexity adds information about curvature under that move, and key-rate durations reveal exposure to selected nonparallel moves. None of these, by itself, establishes that the portfolio is hedged against every possible change in rates.

A disciplined assessment separates the questions being asked:

Check the parallel first-order exposure.
Compare asset and liability dollar durations. A zero net amount indicates local neutrality to a small parallel shift under the duration model.

Check parallel curvature.
Compare asset and liability dollar convexities. A mismatch indicates a second-order effect even when dollar durations match.

Check curve-shape exposures.
Compare key-rate dollar exposures at each chosen maturity and test relevant twists or bends, rather than relying only on the sum.

Reprice material scenarios.
For large shifts, unusual curve shapes, or nonlinear cash flows, revalue assets and liabilities under the complete scenario and compare the resulting surplus.

A common exam trap is to use a duration match as proof of a complete hedge. For example, the wrong conclusion in the curve-twist example is: “The asset and liability durations both total five years, so surplus is unchanged.” That calculation implicitly applies one identical yield change at every maturity. The actual scenario has different changes at two and ten years, so the relevant first-order calculation is $-\sum_j(AK_{A,j}-LK_{L,j})\Delta y_j$, which gives a $2 million increase in the estimate.

Other limitations include the choice of curve nodes and bump shapes, changes in duration and convexity as time passes, and cash flows that respond to rates. If embedded options or policyholder behavior make cash flows rate-dependent, fixed-cash-flow convexity may not capture the full response. In those cases, scenario repricing must reflect the modeled cash-flow behavior. These limitations do not make duration, convexity, or key-rate duration useless; they identify what each measure does and does not cover.

Key takeaway: Duration matching controls a local parallel-shift effect. Convexity measures second-order parallel sensitivity, while key-rate durations expose sensitivity to selected curve movements. A credible immunization assessment states its shift assumptions and uses scenario repricing when those local measures are not sufficient.

Check Your Understanding

For each problem, use the stated local approximation and show the sign of the resulting change. Amounts are hypothetical.

A bond has modified duration 4.0 years and convexity 30 squared years. Estimate its percentage price change when yield rises by 50 basis points. (Answer: $-1.9625\%$)
Assets and liabilities are each worth $80 million and each has duration 6.0 years. Asset convexity is 50 and liability convexity is 35 squared years. Estimate the surplus change for a parallel yield rise of 80 basis points using the second-order approximation. (Answer: surplus increases by $38,400)
Assets and liabilities are each worth $50 million. Asset key-rate durations at two and eight years are 2.0 and 1.0; liability key-rate durations are 1.0 and 2.0. Estimate the first-order surplus change when the two-year yield rises by 100 basis points and the eight-year yield falls by 50 basis points. (Answer: surplus decreases by $750,000)
For a portfolio, $A=L=\$60$ million. Asset key-rate durations at three and seven years are 1.5 and 2.5; liability key-rate durations are 2.0 and 2.0. Estimate the first-order surplus change when the three-year yield falls by 25 basis points and the seven-year yield rises by 25 basis points. (Answer: surplus decreases by $150,000)
An asset-liability portfolio has equal values and equal parallel durations, but the assets have lower convexity than the liabilities. Under the stated second-order approximation, what is the sign of the surplus change for a small nonzero parallel shift, and what important qualification applies? (Answer: the estimated surplus change is negative for either direction of a small parallel shift; this is a local approximation and does not establish the outcome for large shifts or nonparallel scenarios.)

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