Rebalancing a Duration-Matched Portfolio

Why a Duration Match Needs Rebalancing

A duration-matched portfolio is constructed for a particular valuation date, yield environment, liability amount, and set of cash flows. Those inputs can change. A yield change alters market values and may alter durations; a revised liability estimate changes the amount or timing of the obligation; and the passage of time changes the remaining cash-flow schedule. As a result, holdings that matched yesterday may no longer match today.

Rebalancing means recalculating the current liability value and duration, then changing asset holdings so that the portfolio meets the chosen value and duration targets again. The calculation is a local risk-control technique: it aligns asset and liability sensitivity to a small, common yield change at the rebalance date. It does not guarantee exact funding under large or nonparallel yield movements, or when cash flows and assumptions change again.

Definition: A duration-matched portfolio at a valuation date has asset value $A$ equal to the liability value $L$, and asset dollar duration equal to liability dollar duration. If the assets have market values $x_j$ and durations $D_j$, while the liability has value $L$ and duration $D_L$, the two conditions are $$ \sum_j x_j=L \qquad\text{and}\qquad \sum_j x_jD_j=LD_L. $$ For a common valuation yield, the same yield convention must be used for every duration in the comparison.

Here, holdings $x_j$ are market values, not face amounts. The equations are most directly interpreted using Macaulay durations: when all cash flows are discounted at the same effective annual yield, matching value and Macaulay duration also matches first-order dollar sensitivity. Equivalently, one can use modified durations on both sides, since the common factor $1/(1+i)$ converts Macaulay to modified duration.

The Rebalancing Calculation

With two assets, the target holdings are determined by two simultaneous equations. The first preserves the required total market value. The second matches the liability’s dollar duration. Once the target holdings are known, compare them with current market values to determine what to buy and sell.

Update the liability.
At the current valuation date and yield basis, calculate its present value and duration from the current liability cash flows.

Update the assets.
Revalue each asset and recalculate its duration using its remaining cash flows and the current yield basis.

Solve the target equations.
Set target asset value equal to the liability value and target asset dollar duration equal to liability dollar duration.

Calculate trades and verify.
Subtract current market values from target market values. Check both value and duration after the trades.

For two assets with durations $D_1$ and $D_2$, let $x$ be the target value in asset 1 and $y$ the target value in asset 2. If $D_1\ne D_2$, solving the two equations gives

x+y=L,\qquad xD_1+yD_2=LD_L,

and therefore

x=L\frac{D_2-D_L}{D_2-D_1}, \qquad y=L\frac{D_L-D_1}{D_2-D_1}.

These expressions also reveal a feasibility condition when holdings must be nonnegative: $D_L$ must lie between $D_1$ and $D_2$. If it does not, no nonnegative combination of just those two assets can match the liability’s duration and value.

Why Value and Duration Are the Right Local Targets

For cash flows discounted at a common annual effective yield $i$, a price $P$ has Macaulay duration $D$ and modified duration $D/(1+i)$. For a small yield change $\Delta i$, its first-order price change is approximately

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

Thus matching asset and liability values and durations makes their first-order dollar changes equal for the same small yield change. A short derivation shows the role of each condition. Let the surplus be $S=A-L$. At the rebalance date, $A=L$, so the initial surplus is zero. Under a common small yield change, the first-order change in surplus is

\Delta S \approx -\frac{AD_A-LD_L}{1+i}\Delta i.

If the dollar durations match, $AD_A=LD_L$, the coefficient of $\Delta i$ is zero. The surplus therefore has no first-order response to a small common yield change. This is a local statement, not a guarantee of equal values after a large yield movement. Convexity, nonparallel movements, transaction costs, and new cash-flow information can all matter in practice.

Worked Example: Correcting an Overfunded Portfolio

At the current valuation date, a liability has market value $960 and Macaulay duration 4 years. The fund holds $450 of an asset with duration 2 years and $600 of an asset with duration 6 years. Assume the values and durations use the same annual effective yield basis, holdings can be changed without transaction costs, and short sales are not needed. Find the target holdings and the required trades.

Method. Solve the value and dollar-duration equations for the target holdings. Then compare each target with its current value.

Let $x$ be the target market value of the 2-year-duration asset and $y$ the target market value of the 6-year-duration asset. The value equation is

$$ x+y=960. $$

The liability’s dollar duration is $960(4)=3{,}840$ dollar-years. The duration equation is

$$ 2x+6y=3{,}840. $$

Subtract twice the value equation, $2x+2y=1{,}920$, from the duration equation:

4y=1{,}920 \quad\Longrightarrow\quad y=480.

Then $x=960-480=480$. The target is $480 in each asset.

Current holdings are $450 in the shorter-duration asset and $600 in the longer-duration asset. The fund must buy $480-450=$$30 of the shorter-duration asset and sell $600-480=$$120 of the longer-duration asset. The gross sale is $120. The net reduction in total assets is

$$ (450+600)-(480+480)=1{,}050-960=90. $$

Check. Target value is $480+480=960$, equal to the liability value. Target dollar duration is $2(480)+6(480)=960+2{,}880=3{,}840$, equal to the liability’s dollar duration. The $30 purchase and $120 sale produce the correct net reduction of $90.

Common mistake avoided. Net funding change is not the same as gross sales. Selling $120 and buying $30 reduces total assets by $90; it does not reduce them by $30.

Rebalancing After a Yield Change

A yield change requires more than updating the liability value. Asset values and durations also need to be recalculated from the remaining cash flows. For a zero-coupon bond paying face amount $F$ at time $n$, under a single annual effective yield $i$, its value is $Fv^n=F/(1+i)^n$ and its Macaulay duration is $n$ years. The duration is its payment time, but its market value still changes with the yield.

Worked Example: Rebalancing After a Parallel Yield Change

A fund must pay $110.25 in 2 years. It holds two zero-coupon bonds: one pays $55 in 1 year, and the other pays $66.55 in 3 years. Initially the annual effective yield is 10%; it then changes to 5%. Assume all cash flows are certain, the same annual effective yield applies to each maturity, and there are no transaction costs. Revalue the liability and assets at 5%, then find the target asset holdings.

Method. Reprice each cash flow at the new yield, use the zero-coupon payment time as its duration, and solve the two target equations.

At 5%, $v=1/1.05$. The liability value is

L=\frac{110.25}{(1.05)^2} =\frac{110.25}{1.1025} =100.

The liability has a single payment at year 2, so its duration is 2 years and its dollar duration is $100(2)=200$ dollar-years.

The 1-year bond’s new value is

\frac{55}{1.05}=52.380952\ldots.

Its duration is 1 year. The 3-year bond’s new value is

\frac{66.55}{(1.05)^3} =\frac{66.55}{1.157625} =57.488393\ldots.

Its duration is 3 years. Let $x$ and $y$ denote the target values of the 1-year and 3-year bonds. Then

x+y=100,\qquad x+3y=200.

Subtracting the first equation from the second gives $2y=100$, so $y=50$ and $x=50$. At the new yield, the fund therefore sells $52.380952\ldots-50=2.380952\ldots$ of the 1-year bond and $57.488393\ldots-50=7.488393\ldots$ of the 3-year bond.

Check. The target assets total $100, and their dollar duration is $1(50)+3(50)=200$ dollar-years, matching the liability. Rounded to the nearest cent, the fund sells $2.38 and $7.49, for total sales of $9.87. The displayed trade amounts are rounded; the target equations use the unrounded values.

Common mistake avoided. The old 10% values cannot be carried forward after the yield change. Revalue each payment at 5% before finding the trades.

Rebalancing After Liability Cash Flows Change

Even if the yield basis has not changed, an updated liability estimate can shift both value and duration. A revised payment amount, a new obligation, or a change in the timing of payments can alter the target. The correct procedure is to value the current liability cash flows first; applying the old liability duration to a new liability value is generally not valid.

Worked Example: Updating Holdings for a Revised Payment

At an annual effective yield of 5%, a fund currently holds $450 in an asset of duration 2 years and $450 in an asset of duration 6 years. These holdings match an old liability value of $900 and duration 4 years. The liability cash flows are revised to $496.125 at year 2 and $737.05260234375 at year 6. Assume certain payments, the same 5% annual effective yield for all cash flows, and no transaction costs. Find the revised target holdings and trades.

Method. Recalculate the present value of each revised payment and use those present-value components to calculate the liability duration.

The discount factors are based on $(1.05)^2=1.1025$ and $(1.05)^6=1.340095640625$. The year-2 payment has present value

\frac{496.125}{1.1025}=450.

The year-6 payment has present value

\frac{737.05260234375}{1.340095640625}=550.

Thus the revised liability value is $L=450+550=1{,}000$. Its Macaulay duration is the present-value-weighted average payment time:

D_L=\frac{2(450)+6(550)}{1{,}000} =\frac{900+3{,}300}{1{,}000} =4.2\text{ years}.

Let $x$ and $y$ be the target values in the 2-year- and 6-year-duration assets. The equations are

x+y=1{,}000,\qquad 2x+6y=1{,}000(4.2)=4{,}200.

Subtract twice the first equation from the second: $4y=2{,}200$, so $y=550$, and $x=450$. The fund keeps $450 in the shorter-duration asset and buys $550-450=$$100 of the longer-duration asset.

Check. The target value is $450+550=1{,}000$, and target dollar duration is $2(450)+6(550)=900+3{,}300=4{,}200$ dollar-years. Both match the revised liability. The previous holdings matched the old value and duration, but not this updated liability.

Common mistake avoided. The new liability value is $1,000, but its duration is not automatically the old 4 years. It must be recalculated using the present values and times of the revised cash flows.

Practical Limits and Common Traps

A duration match is date-specific. As time passes, remaining times to payment change, and asset and liability values may evolve differently. A duration match also depends on the valuation assumptions: if the asset and liability calculations use inconsistent yield conventions, the weighted-duration equations do not represent a valid sensitivity comparison. In a practical review, document the valuation date, yield basis, cash flows, values, durations, and trades.

Do not use stale durations. Recalculate duration when yields or remaining cash flows change.
Separate the value and duration tests. A portfolio can match total value without matching dollar duration, or match duration while having too little or too much value.
Track purchases and sales separately from net funding. The total of asset sales less purchases gives the net cash released; it is not necessarily the gross amount sold.
Check nonnegative holdings. If the liability duration is outside the range of the available asset durations, a two-asset match requires a negative holding in one asset. If short sales are prohibited, another asset or a different target is needed.
Do not treat first-order protection as exact matching. The duration equations control local sensitivity to a small common yield change. They do not replicate cash flows or eliminate every source of surplus risk.

The central discipline is to rebalance from current values and current cash flows, not from the original portfolio design. Recompute the liability target, revalue the available assets, solve both matching equations, and verify the proposed trades against both conditions.

Check Your Understanding

In each question, treat stated durations as Macaulay durations at the current valuation date and assume no transaction costs unless noted.

A liability is valued at $840 with duration 3 years. Two assets have durations 1 and 5 years. Find the target market value in each asset. (Answer: $420 in each asset.)
A liability’s value rises to $900 and its recalculated duration is 4 years. Available assets have durations 2 and 6 years. Find target holdings. (Answer: $450 in each asset.)
At an annual effective yield of 5%, a liability consists of a $220.50 payment in 2 years and a $737.05260234375 payment in 6 years. Assets have durations 2 and 6 years. Find the liability value, duration, and target holdings. (Answer: Value $750; duration $4.9333$ years; $200 in the duration-2 asset and $550 in the duration-6 asset.)
A liability has value $900 and duration 3.5 years. Current holdings are $300 in an asset of duration 1 year and $600 in an asset of duration 5 years. Find the target holdings and trades. (Answer: Target $337.50 in the duration-1 asset and $562.50 in the duration-5 asset; buy $37.50 of the first and sell $37.50 of the second.)
At an annual effective yield of 4%, a liability pays $108.16 in 2 years. Two zero-coupon assets pay $104 in 1 year and $112.4864 in 3 years. Revalue all three cash flows and find the duration-matched target values. (Answer: Liability value $100 and duration 2 years; each asset value is $100; target $50 in each asset.)

What You'll Learn