Duration Matching and First-Order Neutrality

Why Duration Matching Matters

Matching an asset portfolio’s value to a liability’s value at a chosen valuation rate is not enough to make the two values respond alike when yields move. If the assets and liabilities have different durations, a small yield change generally changes their present values by different amounts. Duration matching addresses that first-order sensitivity: when present values and durations match, the first-order change in surplus is zero for a small parallel shift in yield.

The result is local. It concerns the linear, first-order effect of a small shift from the valuation rate, not a guarantee that surplus remains unchanged after a larger shift. The discussion here assumes fixed, deterministic cash flows and a common annual effective yield applied to every payment. Those assumptions make the sensitivity calculation precise and are important when deciding whether it applies.

Definition: At annual effective yield $i$, the present value of fixed cash flows $C_t$, paid at times $t$ years from now, is $P(i)=\sum_t C_t(1+i)^{-t}$. The Macaulay duration is $D(i)=\frac{\sum_t tC_t(1+i)^{-t}}{P(i)}$. It is the present-value-weighted average payment time. Write $P_A,D_A$ for the asset present value and duration, and $P_L,D_L$ for the liability present value and duration, all measured at the same valuation rate.

From Duration to First-Order Sensitivity

For a portfolio with fixed cash flows, duration summarizes the first-order response of present value to a change in the annual effective yield. If the yield changes from $i$ to $i+\Delta i$, the first-order approximation is

\Delta P \approx -\frac{P(i)D(i)}{1+i}\Delta i.

The factor $1/(1+i)$ is needed because the formula uses Macaulay duration and an annual effective yield. The corresponding modified duration is $D(i)/(1+i)$; it is modified duration, not Macaulay duration by itself, that gives the approximate percentage price change per unit change in yield.

Why the Sensitivity Formula Holds

For a payment $C_t$ at time $t$, its present value at yield $i$ is $C_t(1+i)^{-t}$. Differentiating with respect to $i$ gives

\frac{d}{di}\left[C_t(1+i)^{-t}\right] =-tC_t(1+i)^{-t-1}.

Summing over the fixed cash flows and factoring out $1/(1+i)$ gives

\frac{dP}{di} =-\frac{1}{1+i}\sum_t tC_t(1+i)^{-t} =-\frac{P(i)D(i)}{1+i}.

For a small yield change, the first-order change is the derivative at the valuation rate multiplied by $\Delta i$. This yields the approximation above. It does not state that the price change is exactly linear in the yield change; it omits higher-order effects.

The First-Order Neutrality Condition

Define surplus as asset value less liability value, $S=P_A-P_L$. A positive surplus means that the assets exceed the liabilities at the valuation rate. For the fixed-cash-flow model, the asset and liability values are both affected by the same parallel yield shift, so subtracting their first-order changes gives

\Delta S \approx -\frac{P_A D_A-P_L D_L}{1+i}\Delta i.

Thus first-order neutrality requires $P_A D_A=P_L D_L$. If the present values are also matched, $P_A=P_L$, that condition reduces to duration matching, $D_A=D_L$. The two matching conditions work together: equal present values alone do not ensure equal sensitivity, and equal durations alone do not ensure equal sensitivity if present values differ.

Theorem: Under a common annual effective yield, with fixed deterministic asset and liability cash flows, if $P_A=P_L$ and $D_A=D_L$ at the valuation rate $i$, then the first-order change in surplus from a small parallel yield shift $\Delta i$ is zero. This is local first-order neutrality, not exact neutrality for every yield shift.

Worked Applications

Worked Example: Allocating Value Between Two Zero-Coupon Assets

A liability pays $1,331 at the end of year 3. The valuation rate is an annual effective rate of $10\%$. Two available zero-coupon assets pay at the ends of years 2 and 4. Find the present value to assign to each asset so that the asset portfolio matches the liability’s present value and duration.

Method. First value the liability and determine its duration. Then solve the two matching equations for the present values assigned to the assets.

At $i=0.10$, the liability present value is

P_L=\frac{1331}{(1.10)^3} =\frac{1331}{1.331} =1000.

There is one liability payment, at year 3, so its Macaulay duration is $D_L=3$ years. Let $x$ be the present value assigned to the year-2 asset and $y$ the present value assigned to the year-4 asset. Present-value matching requires $x+y=1000$. Duration matching requires

\frac{2x+4y}{x+y}=3, \qquad 2x+4y=3000.

Subtracting $2x+2y=2000$, which follows from the value equation, gives $2y=1000$, so $y=500$ and $x=500$. The required present-value allocation is therefore $500 to each asset.

Check. The asset values total $500+500=1000$, and their duration is $(2(500)+4(500))/1000=3$ years. The liability’s maturity payment is also consistent with the valuation: $1000(1.10)^3=1331$. Both matching conditions hold, so the first-order surplus change is zero for a small parallel shift.

Worked Example: Matching Duration from a Multi-Payment Liability

At an annual effective valuation rate of $5\%$, a liability pays $126 at the end of year 1 and $208.3725 at the end of year 3. The available assets are zero-coupon instruments maturing at the ends of years 1 and 4. Find asset present values that match both the liability value and its duration.

Method. Calculate the present value of each liability payment first. These present values are the weights used to calculate the liability’s duration.

The year-1 payment has present value $126/1.05=120$. Since $(1.05)^3=1.157625$, the year-3 payment has present value $208.3725/1.157625=180$. Therefore,

P_L=120+180=300, \qquad D_L=\frac{1(120)+3(180)}{300} =\frac{660}{300} =2.2\text{ years}.

Let $x$ be the present value of the year-1 asset and $y$ the present value of the year-4 asset. Value matching gives $x+y=300$. Duration matching gives $x+4y=2.2(300)=660$. Subtracting the value equation from the duration equation yields $3y=360$, so $y=120$ and $x=180$.

Check. The asset present values sum to $180+120=300$, and their weighted duration is $(1(180)+4(120))/300=660/300=2.2$ years. At $5\%$, the corresponding asset payments are $180(1.05)=189$ at year 1 and $120(1.05)^4=145.86075$ at year 4. Discounting them recovers present values of $180 and $120, respectively.

The common mistake avoided here is averaging payment times using the cash-flow amounts instead of their present values. Duration weights each payment by its value at the valuation rate.

Worked Example: Measuring the Effect of a Duration Mismatch

An asset portfolio and a liability portfolio each have a present value of $600,000 at an annual effective yield of $5\%$. Their durations are 4.0 years and 3.5 years, respectively. Estimate the first-order change in surplus if the yield rises by 0.20 percentage points.

Method. Use the surplus-sensitivity formula. A rise of 0.20 percentage points is an absolute rate change of $\Delta i=0.002$, not $0.20$.

The difference in present-value-duration products is

P_A D_A-P_LD_L =600{,}000(4.0)-600{,}000(3.5) =2{,}400{,}000-2{,}100{,}000 =300{,}000.

Consequently, the first-order surplus change is

\Delta S\approx -\frac{300{,}000}{1.05}(0.002) =-571.42857\ldots

Rounded to the nearest cent, surplus is estimated to decrease by $571.43. The negative sign is consistent with the assets having the larger duration: for the same matched present value, they are more sensitive to a yield increase.

If the portfolio could instead be allocated between assets with durations 2 and 5 years, let their present values be $x$ and $y$. Matching the liability requires $x+y=600{,}000$ and $2x+5y=3.5(600{,}000)=2{,}100{,}000$. Subtracting twice the first equation from the second gives $3y=900{,}000$, so $y=300{,}000$ and $x=300{,}000$. This allocation has duration $(2(300{,}000)+5(300{,}000))/600{,}000=3.5$, matching the liability.

Common Errors and Limits

The matching conditions are simple to state, but they are easy to apply incorrectly. In particular, duration is a weighted average, and the weights must be present values at the same valuation rate. The yield change must also be entered as a decimal change in the annual effective rate.

Common mistake: Suppose a liability has present value $800,000 and duration 3.5 years, and assets have durations 2 and 5 years. Assigning $533,333.33 to the duration-2 asset and $266,666.67 to the duration-5 asset matches the total value, but not duration: $(2(533{,}333.33)+5(266{,}666.67))/800{,}000=3.0$, not 3.5. The correct equations are $x+y=800{,}000$ and $2x+5y=3.5(800{,}000)=2{,}800{,}000$. Solving gives $x=y=400{,}000$, whose weighted duration is 3.5 years.

Matching only present values. Equal values do not imply equal first-order sensitivity when durations differ.
Matching only durations. If present values differ, equal durations do not generally make $P_A D_A=P_LD_L$.
Using the wrong yield change. A 0.20 percentage-point increase is $0.002$, not $0.20$.
Treating the approximation as exact. First-order neutrality does not rule out a nonzero surplus change from higher-order effects or a large yield movement.

The result also relies on the specified model. A single parallel shift applies the same yield change to every cash flow. If yields at different maturities move by different amounts, one aggregate duration generally does not capture all the effects. Likewise, if cash flows change when rates change—for example, because an instrument can be called—fixed-cash-flow sensitivity is not by itself a complete repricing model. For deterministic fixed cash flows and a small parallel shift, duration matching is a direct and useful first-order test, aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated annual effective valuation rate and treat the yield movement as a parallel shift.

A liability has present value $500,000 and duration 4 years. Two zero-coupon assets have durations 2 and 6 years. What present values should be assigned to match value and duration? (Answer: $250,000 to each asset)
At $i=4\%$, a liability has present-value components of $100 at year 1 and $200 at year 3. Find its present value and duration. (Answer: Present value $300; duration $7/3$ years, or approximately 2.3333 years)
Using the liability in question 2, assets mature at years 1 and 4. What present values should be assigned to match both value and duration? (Answer: $166.67 at year 1 and $133.33 at year 4, to the nearest cent)
Assets and liabilities each have present value $1,000,000 at $i=4\%$. Their durations are 3.8 and 3.2 years, respectively. Estimate the first-order surplus change for a yield decrease of 0.10 percentage points. (Answer: Increase of approximately $576.92)
Asset present value is $1,020,000 with duration 5 years; liability present value is $1,000,000 with duration 5 years. At $i=5\%$, estimate the first-order surplus change for a yield increase of 0.10 percentage points. (Answer: Decrease of approximately $95.24)

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