Deriving First-Order Duration Matching

Why First-Order Duration Matching Works

An asset-liability portfolio is exposed to interest-rate changes because the present values of its future cash flows change when yields change. Duration measures the first-order sensitivity of present value to a yield change. The key question is not simply whether the assets and liabilities have the same duration: it is whether their dollar sensitivities agree, so that the change in asset value offsets the change in liability value.

This tutorial derives that condition from the present-value formula. The derivation assumes fixed cash flows and a small, parallel change in one common annual effective yield. It describes local, first-order behavior; it does not assert that surplus remains unchanged for a large rate move.

Definition: Let $C_t$ be a fixed cash flow at time $t$, measured in years, and let $y$ be the common annual effective yield used to discount all cash flows. Its present value is $V(y)=\sum_t C_t(1+y)^{-t}$. The Macaulay duration is $D_M=\sum_t tC_t(1+y)^{-t}/V(y)$, and the modified duration is $D_{\mathrm{mod}}=D_M/(1+y)$. The dollar modified duration is $V D_{\mathrm{mod}}$. For liabilities, $V$ denotes the positive present value of the liability amount, not a negative cash flow.

For a portfolio of assets, write its present value and Macaulay duration as $A$ and $D_A$. For a liability portfolio, write its positive present value and Macaulay duration as $L$ and $D_L$. Define surplus as $S=A-L$. In this common-yield model, a small yield change $\Delta y$ changes each present value approximately in proportion to its dollar modified duration. The asset value usually falls when yields rise; the liability value usually falls as well. Surplus depends on the difference between those changes.

Deriving the First-Order Condition

Begin with fixed cash flows $C_t$ discounted at the common effective yield $y$. Differentiating each discounted cash flow gives a factor of $-t/(1+y)$. Summing the derivatives and using the definition of Macaulay duration yields the present-value sensitivity.

Differentiate each discounted cash flow.
For a fixed amount $C_t$, the derivative of $C_t(1+y)^{-t}$ with respect to $y$ is $-tC_t(1+y)^{-t-1}$.

Sum across payment dates.
The derivative of portfolio present value is the sum of those individual derivatives.

Use Macaulay duration.
The time-weighted present-value sum equals $V D_M$, which gives the modified-duration form.

Apply it to surplus.
Subtract the liability change from the asset change. The first-order surplus change is zero exactly when the two dollar modified durations are equal.

\begin{aligned} \frac{dV}{dy} &= -\sum_t \frac{tC_t}{(1+y)^{t+1}} \\ &= -\frac{1}{1+y}\sum_t t\frac{C_t}{(1+y)^t} \\ &= -\frac{V D_M}{1+y} = -V D_{\mathrm{mod}}. \end{aligned}

For a small change $\Delta y$, the first-order approximation is $\Delta V\approx -V D_{\mathrm{mod}}\Delta y$. Applying this to assets and liabilities separately, and recalling that surplus is asset value minus liability value, gives the general condition.

\Delta S \approx \left(-A\frac{D_A}{1+y} +L\frac{D_L}{1+y}\right)\Delta y.

Theorem: Under the common-yield, fixed-cash-flow model, surplus has zero first-order change for a small parallel yield shift if and only if $A D_A=L D_L$. Equivalently, the asset and liability dollar modified durations are equal: $A D_A/(1+y)=L D_L/(1+y)$. If present values are also matched, so $A=L>0$, this condition reduces to $D_A=D_L$.

The equivalence follows because both sides use the same yield, so division by $1+y$ does not change whether they are equal. The familiar requirement of equal durations therefore relies on the separate present-value matching condition. If the present values differ, equal Macaulay durations are not generally the right test; compare the value-weighted durations instead.

Worked Example: Deriving and Checking a Matched Portfolio

At a common annual effective yield of 10%, a hypothetical asset portfolio pays $155 at year 1 and $66.55 at year 3. A liability portfolio requires $100 at year 1 and $121 at year 2. Determine whether present values and durations match, then estimate the first-order surplus change if the yield rises by 1 percentage point.

Method. Discount each cash flow at 10%, calculate the present-value-weighted payment time for each side, and compare dollar modified durations. This avoids assuming that matching payment dates or cash-flow totals alone determine interest sensitivity.

Asset present value. The year-1 cash flow has present value $155/1.10=\$140.9091$. Since $1.10^3=1.331$, the year-3 cash flow has present value $66.55/1.331=\$50.0000$. Thus $A=140.9091+50.0000=\$190.9091$.

Liability present value. The year-1 payment has present value $100/1.10=\$90.9091$. The year-2 payment has present value $121/1.10^2=121/1.21=\$100.0000$. Thus $L=90.9091+100.0000=\$190.9091$. Present values match.

Durations. The asset’s time-weighted present-value sum is $1(140.9091)+3(50.0000)=\$290.9091$, so $D_A=290.9091/190.9091=1.52381$ years. The liability’s sum is $1(90.9091)+2(100.0000)=\$290.9091$, so $D_L=290.9091/190.9091=1.52381$ years. Durations match as well.

The common dollar modified duration is $190.9091(1.52381)/1.10=\$264.4628$ per unit yield. For $\Delta y=0.01$, the first-order change in each value is approximately $-264.4628(0.01)=-\$2.6446$. The surplus change is therefore approximately $-2.6446-(-2.6446)=\$0.00$. The check is that both value changes agree, consistent with equal present values and durations. This is a first-order estimate, not a claim that the exact change is zero.

When Present Values Do Not Match

The general test is equality of dollar modified durations, not equality of Macaulay durations by themselves. It is possible for the first-order change in surplus to be zero even when the asset and liability present values differ. In that case, however, the surplus itself is not zero: the condition only says its local rate sensitivity is zero.

Worked Example: Equal Present Values but Different Durations

At a 5% annual effective yield, an asset pays $330.75 at year 2, while a liability requires $347.2875 at year 3. Estimate the first-order surplus change for a 1-percentage-point yield increase.

Method. Calculate each present value and duration from its dated cash flow, then apply the surplus derivative. The common mistake to avoid is treating equal present values as sufficient for first-order matching.

The asset present value is $330.75/1.05^2=330.75/1.1025=\$300$, and its Macaulay duration is $D_A=2$ years. The liability present value is $347.2875/1.05^3=347.2875/1.157625=\$300$, and its duration is $D_L=3$ years. The present values match, but the durations do not.

The asset dollar modified duration is $300(2)/1.05=\$571.4286$ per unit yield. The liability dollar modified duration is $300(3)/1.05=\$857.1429$. For $\Delta y=0.01$, the first-order asset change is approximately $-571.4286(0.01)=-\$5.7143$, and the liability change is approximately $-857.1429(0.01)=-\$8.5714$. Since surplus subtracts the liability change, $\Delta S\approx-5.7143-(-8.5714)=+\$2.8571$. A rise in yield therefore increases surplus to first order. The result checks against the general formula: liability dollar duration exceeds asset dollar duration.

Worked Example: Matching Dollar Duration Without Matching Present Value

At a 5% annual effective yield, a hypothetical asset pays $661.50 at year 2 and a liability requires $463.05 at year 3. Find the first-order surplus change for a 1-percentage-point yield increase, and state whether the starting surplus is zero.

Method. Check the present values, then compare value-weighted durations. This example tests the general condition rather than the special case in which present values match.

The asset present value is $661.50/1.05^2=661.50/1.1025=\$600$, with $D_A=2$ years. The liability present value is $463.05/1.05^3=463.05/1.157625=\$400$, with $D_L=3$ years. Their present values do not match, so starting surplus is $S=600-400=\$200$.

The asset dollar modified duration is $600(2)/1.05=\$1,142.8571$ per unit yield. The liability dollar modified duration is $400(3)/1.05=\$1,142.8571$ as well. For $\Delta y=0.01$, each value changes by approximately $-1,142.8571(0.01)=-\$11.4286$. Thus $\Delta S\approx-11.4286-(-11.4286)=\$0.00$. The first-order condition holds, but the portfolio is not initially value-matched: its surplus is $200. The calculation checks that zero first-order change and zero starting surplus are distinct statements.

Assumptions, Limits, and Exam Traps

The derivative above applies to fixed dated cash flows discounted using one common annual effective yield. The yield shift is additive, measured in the same annual rate units as $y$, and small enough that a first-order approximation is appropriate. For example, a rise from 5% to 5.5% is $\Delta y=0.005$, not $0.5$. The formula does not by itself describe cash flows that change with rates, a nonparallel movement in separate spot rates, or the exact effect of a large yield shock.

For a portfolio with matched present values, equal Macaulay durations imply equal dollar modified durations, and hence zero first-order surplus change. Without present-value matching, the implication fails. Example 3 shows why: the asset and liability durations are 2 and 3 years, but their present values weight those durations so that $A D_A=L D_L$.

Common exam trap: Concluding that equal durations always imply first-order matching, or that zero first-order surplus change requires equal durations. The correct test is $A D_A=L D_L$. Equal durations are sufficient when $A=L$; when values differ, compare the products. Also keep the sign convention consistent: $\Delta S=\Delta A-\Delta L$, not $\Delta A+\Delta L$.

This condition is useful as a local diagnostic for whether an asset portfolio offsets liability sensitivity to a small parallel yield shift. It does not establish exact cash-flow matching, eliminate the initial surplus or deficit, or guarantee stability for larger or nonparallel rate movements. The distinction is important: first-order duration matching addresses the slope of surplus with respect to yield at the valuation point, not every aspect of asset-liability risk. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated common annual effective yield and interpret the requested change as a first-order estimate.

At a 4% yield, assets have present value $250 and Macaulay duration 4 years. Liabilities have present value $250 and duration 3.2 years. Estimate the surplus change for a yield increase of 0.005. (Answer: approximately $-\$0.96$.)
At a 6% yield, assets and liabilities each have present value $800. Their durations are 5 years and 4.5 years, respectively. Estimate the surplus change if yield falls by 0.002. (Answer: approximately $+\$0.75$.)
At a 5% yield, assets have present value $900 and duration 3 years; liabilities have present value $600 and duration 4.5 years. What is the first-order surplus change for a 0.01 yield increase, and what is the initial surplus? (Answer: approximately $0.00 change; initial surplus is $300.)
At a common yield, asset present value is $720 and asset duration is 2.8 years. Liability present value is $630. What liability duration makes first-order surplus change zero? (Answer: 3.2 years.)
At a common yield, asset dollar modified duration is $1,600 per unit yield, and liability dollar modified duration is also $1,600 per unit yield. Estimate each side’s value change and the surplus change for a yield increase of 0.003. (Answer: each value decreases by approximately $4.80; surplus changes by approximately $0.00.)

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