Duration Matching for Project Liabilities

Why Match Project Liabilities?

A project can have an obligation due in the future even when its assets are already in place. Matching the assets’ value to the liability’s present value is a useful starting point, but it does not by itself control the effect of changing yields. If the assets and liability respond differently to a rate change, the project’s surplus can move even though it began with zero surplus.

Duration matching addresses this first-order rate sensitivity. It is a local approximation: when asset and liability present values and durations match, a small parallel yield shift has approximately no first-order effect on surplus, under the common-yield and fixed-cash-flow assumptions below. It does not guarantee that the surplus remains unchanged for larger shifts.

Definition: For fixed positive cash flows $C_t$ at times $t=1,\ldots,n$, valued at a common effective annual yield $i$, the present value is $V(i)=\sum_{t=1}^{n}C_t(1+i)^{-t}$. The Macaulay duration is the present-value-weighted average payment time, $D_{\mathrm{Mac}}=\frac{\sum tC_t(1+i)^{-t}}{V(i)}$. The modified duration is $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+i)$, or equivalently $D_{\mathrm{mod}}=-V(i)^{-1}\,dV/di$.

For this tutorial, asset and liability cash flows are fixed, denominated in the same currency, and valued using the same flat effective annual yield. A parallel shift means that this yield changes by the same amount for every cash flow maturity. The approximation is for a small change $\Delta i$; it ignores effects beyond the first-order change. If cash flows are contingent on rates, or assets and liabilities are valued using different yield conventions, these formulas cannot be applied without adjusting the valuation model.

From Duration to Surplus Sensitivity

Let $P_A$ and $P_L$ denote asset and liability present values. Define surplus as $S=P_A-P_L$. For either a fixed asset portfolio or a fixed liability cash-flow stream, the modified-duration approximation is $\Delta P\approx-D_{\mathrm{mod}}P\,\Delta i$. The minus sign matters: a small increase in yield decreases the present value of positive fixed cash flows.

\Delta S \approx -D_A P_A\,\Delta i+D_L P_L\,\Delta i = (D_LP_L-D_AP_A)\Delta i

Here $D_A$ and $D_L$ are modified durations. If present values match, $P_A=P_L=P$, then the first-order surplus change becomes $P(D_L-D_A)\Delta i$. Thus, matching present value and modified duration makes the first-order change zero. Since both sides are valued using the same yield, matching Macaulay durations instead is equivalent: dividing both durations by the same $1+i$ preserves equality.

Why the Modified-Duration Formula Holds

The result follows by differentiating the present-value function. The cash flows are fixed, so each term $C_t(1+i)^{-t}$ has derivative $-tC_t(1+i)^{-t-1}$. Summing these derivatives and dividing by present value gives the relative rate of change.

\frac{dV}{di} =-\sum_{t=1}^{n}\frac{tC_t}{(1+i)^{t+1}} =-\frac{1}{1+i}\sum_{t=1}^{n}\frac{tC_t}{(1+i)^t} =-V(i)\frac{D_{\mathrm{Mac}}}{1+i} =-V(i)D_{\mathrm{mod}}.

For a small change $\Delta i$, the first-order change is $dV\approx (dV/di)\Delta i$. Substituting the derivative gives $\Delta V\approx-D_{\mathrm{mod}}V\Delta i$. Applying this separately to assets and liabilities and subtracting the liability change gives the surplus formula above.

Constructing a Duration-Matched Portfolio

When candidate assets have different maturities, their present values can be allocated to make the portfolio’s duration equal the liability’s duration. For fixed cash flows valued at a common yield, the portfolio duration is the present-value-weighted average of the component durations. The weights are present values divided by total portfolio present value, not the assets’ face amounts.

Value the liability.
Discount each fixed payment at the stated yield. Record both total present value and the present value associated with each payment date.

Calculate its duration.
Use the present-value-weighted payment times for Macaulay duration, or divide by $1+i$ to obtain modified duration.

Set the asset constraints.
Require the asset present value to equal the liability present value and the asset’s present-value-weighted duration to equal the liability’s duration.

Check the first-order surplus change.
Substitute both present values and modified durations into $\Delta S\approx(D_LP_L-D_AP_A)\Delta i$. A matched portfolio gives zero under the stated model.

Worked Example: Matching a Liability with Two Zero-Coupon Assets

A project has fixed obligations of $1,815 at the end of year 2 and $1,996.50 at the end of year 3. Assets can be invested in zero-coupon instruments that pay at the end of year 1 and year 4. All cash flows are valued at a flat effective annual yield of 10%. Find the asset positions, stated as present values, that match the liability’s present value and duration.

Method. Discount each obligation, calculate the liability’s present-value-weighted time, then solve the two asset constraints. This method avoids treating face amounts as duration weights.

Liability present value. The year-2 discount factor is $1/(1.10)^2=1/1.21$, and the year-3 factor is $1/(1.10)^3=1/1.331$. Therefore:

P_L=\frac{1815}{1.21}+\frac{1996.50}{1.331} =1500+1500=3000.

The liability’s Macaulay duration is $(2(1500)+3(1500))/3000=2.5$ years. Its modified duration is $2.5/1.10=2.2727$ years.

Asset positions. Let $x$ and $y$ be the present values invested in the year-1 and year-4 zero-coupon assets. Matching present value requires $x+y=3000$. Matching Macaulay duration requires $x+4y=2.5(3000)=7500$. Subtracting the first equation from the second gives $3y=4500$, so $y=1500$ and $x=1500$.

The corresponding asset payments are $1500(1.10)=1650$ at year 1 and $1500(1.10)^4=2196.15$ at year 4. The asset present value is $1500+1500=3000$, and its Macaulay duration is $(1500+4(1500))/3000=2.5$ years. Its modified duration is $2.5/1.10=2.2727$ years, equal to the liability’s.

Check. Both present values are $3,000, and both modified durations are 2.2727 years. The first-order surplus change is therefore zero for a small parallel yield shift. The assets do not copy the liability payment dates; they match its value and first-order sensitivity.

Matching a More General Asset Portfolio

The same constraints apply when the candidate assets are described by their present values and durations rather than as zero-coupon instruments. For two assets with present values $x$ and $y$, and Macaulay durations $D_1$ and $D_2$, matching a liability with present value $P_L$ and duration $D_L$ requires $x+y=P_L$ and $D_1x+D_2y=D_LP_L$. These equations solve for present-value allocations; the cash invested in each security must then be chosen to produce those present values at the valuation yield.

Worked Example: Solving for Present-Value Allocations

A project liability has present value $10,000 and Macaulay duration 3.5 years at a flat effective annual yield of 5%. Two available zero-coupon assets mature in 1 year and 5 years. Determine the present value to allocate to each asset and state the implied payments.

Method. Solve the present-value and duration equations for the two asset allocations. Using Macaulay duration here is convenient because each zero-coupon asset’s duration is its maturity.

Let $x$ be the present value invested in the 1-year asset and $y$ the present value invested in the 5-year asset. The matching equations are $x+y=10000$ and $x+5y=3.5(10000)=35000$. Subtracting the first equation from the second gives $4y=25000$, so $y=6250$ and $x=3750$.

The 1-year asset pays $3750(1.05)=\$3,937.50$. The 5-year asset pays $6250(1.05)^5=6250(1.2762815625)=\$7,976.76$, rounded to cents.

Check. Asset present value is $3750+6250=\$10,000$, and its Macaulay duration is $(3750(1)+6250(5))/10000=35000/10000=3.5$ years. Its modified duration equals the liability’s because both divide by $1.05$. The portfolio matches both constraints.

Estimating the Surplus Change When Durations Differ

A portfolio may match present value but not duration. In that case the first-order formula estimates the direction and approximate size of the change. A useful sign check is to ask what happens when yields rise: the longer-duration side loses more value per dollar of present value. If assets have the longer duration, surplus tends to fall; if liabilities have the longer duration, surplus tends to rise.

Worked Example: A Small Yield Increase with a Duration Mismatch

At a flat effective annual yield of 4%, a project has assets and liabilities each valued at $12,000. The assets have Macaulay duration 2.5 years, while the liabilities have Macaulay duration 3 years. Estimate the change in surplus for a parallel yield increase of 0.25 percentage points, from 4% to 4.25%.

Method. Convert both Macaulay durations to modified durations, then apply the first-order surplus formula. This avoids using Macaulay duration directly in a yield-change approximation.

The modified durations are $D_A=2.5/1.04=2.403846$ and $D_L=3/1.04=2.884615$. The yield change is $\Delta i=0.0425-0.04=0.0025$, not $0.25$.

\Delta S\approx(2.884615(12000)-2.403846(12000))(0.0025) =14.4231.

Equivalently, the asset-value change is approximately $-2.403846(12000)(0.0025)=-\$72.12$, while the liability-value change is approximately $-2.884615(12000)(0.0025)=-\$86.54$. Since surplus is assets minus liabilities, its estimated change is $-72.12-(-86.54)=+\$14.42$, rounded to cents.

Check. The liability has the longer duration, so its present value falls more than the asset value when yields rise. The surplus therefore increases, consistent with the positive estimate. This is a first-order estimate, not an exact repricing.

Common Mistakes and Scope

Duration matching is a precise first-order condition, not a guarantee that every interest-rate outcome leaves the project’s surplus unchanged. It is useful for organizing a hedge, evaluating a liability-funded investment, and identifying which side of a project is more rate-sensitive. For a nonflat term structure, a parallel shift must be applied consistently to the maturity-specific yields, and sensitivity must be calculated under that valuation model; a single flat-yield duration calculation should not be used automatically.

Reversing the surplus sign. Surplus is $P_A-P_L$. Its change is $\Delta P_A-\Delta P_L$, so the liability term enters with the opposite sign from its own price change.
Using a percentage-point change as a decimal rate. A rise of 0.25 percentage points is $\Delta i=0.0025$, not $0.25$.
Confusing Macaulay and modified duration. Under the common flat effective annual yield, the price-change approximation uses modified duration. Macaulay duration must be divided by $1+i$.
Weighting by face amount. Portfolio duration uses present-value weights. Face amounts are not the appropriate weights unless they happen to be proportional to present values.
Assuming first-order matching is exact. The approximation concerns small parallel shifts and fixed cash flows. It does not establish exact price equality after a finite yield change.

Key takeaway: First match asset and liability present values. Then match their modified durations under the same valuation yield. The resulting surplus has zero first-order sensitivity to a small parallel yield shift; if durations differ, $(D_LP_L-D_AP_A)\Delta i$ gives the approximate surplus change.

Check Your Understanding

For each problem, use fixed cash flows and the stated common flat effective annual yield. Answers involving yield shifts are first-order estimates.

A liability has present-value contributions of $2,400 at year 1 and $1,600 at year 4, valued at 6%. Find its Macaulay and modified durations. (Answer: Macaulay duration $=2.2000$ years; modified duration $=2.0755$ years.)
A liability has present value $8,000 and Macaulay duration 3 years. Find the present-value allocations to zero-coupon assets maturing at years 1 and 5 that match it. (Answer: $4,000 at year 1 and $4,000 at year 5.)
Assets have present value $250,000 and modified duration 4.2 years; liabilities have present value $240,000 and modified duration 3.5 years. Estimate the change in surplus when the yield falls by 0.10 percentage points. (Answer: surplus increases by approximately $210.)
At a 6% yield, assets and liabilities each have present value $12,000. Their Macaulay durations are 2.5 and 2.75 years, respectively. Estimate the surplus change for a yield increase of 0.10 percentage points. (Answer: surplus increases by approximately $2.83.)
A liability has present value $30,000 and Macaulay duration 4 years. Find the present-value allocations to zero-coupon assets maturing at years 2 and 6 that match both quantities. (Answer: $15,000 at year 2 and $15,000 at year 6.)

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