Worked Duration-Matching Calculation

What a Two-Asset Duration Match Does

A duration match uses two assets to align a portfolio with a liability in two respects at a chosen valuation yield: present value and interest-rate sensitivity to first order. The first condition matches the amount of assets to the amount owed. The second matches how quickly those present values are estimated to change when a common yield moves slightly.

This is a local calculation, not a guarantee that assets and liabilities will remain equal for every yield change. The method assumes fixed cash flows and a small, parallel change in the annual effective yield. It is most useful when a liability’s present value and duration are known and two candidate assets have different durations.

Definition: At a stated yield, a two-asset duration match chooses asset market values $V_1$ and $V_2$ so that their total present value equals the liability present value $L$, and their present-value-weighted Macaulay duration equals the liability duration $D_L$: $$ V_1+V_2=L,\qquad V_1D_1+V_2D_2=LD_L. $$ Here $D_1$, $D_2$, and $D_L$ are measured in years. The values $V_1$ and $V_2$ are market values, not necessarily face amounts or numbers of bonds.

Set Up the Match Before Calculating Holdings

For assets with positive present values, the portfolio duration is the weighted average of their durations, with market values as weights. The two equations in the definition can therefore be solved for the required asset market values. If the asset durations differ, the solution is

V_1=L\frac{D_2-D_L}{D_2-D_1}, \qquad V_2=L\frac{D_L-D_1}{D_2-D_1}.

These expressions are valid when $D_1\ne D_2$. A solution with both values positive requires the liability duration to lie between the asset durations. If it does not, the equations still have an algebraic solution when the durations differ, but at least one holding will be negative; whether short holdings are allowed is a separate constraint.

Value each asset at the stated yield.
Use the asset’s dated cash flows and the annual effective yield to determine its market value and duration.

Solve for market-value allocations.
Use the present-value equation and the duration-weighted equation together.

Translate allocations into holdings.
Divide an allocation by the bond’s price per unit, or discount a zero-coupon face amount to find its present value.

Check both conditions.
Recalculate the total present value and the portfolio’s weighted duration from the chosen holdings.

Why the Match Cancels First-Order Yield Changes

For fixed cash flows $C_t$ paid at times $t$, valued at annual effective yield $i$, present value is $P(i)=\sum_t C_t(1+i)^{-t}$. Macaulay duration is the present-value-weighted average payment time:

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

Differentiate the present-value expression with respect to yield. Each term has derivative $-tC_t(1+i)^{-t-1}$. Factoring out $-1/(1+i)$ gives

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

Thus modified duration is $D/(1+i)$, and for a small yield change $\Delta i$, the first-order estimate is $\Delta P\approx-PD\Delta i/(1+i)$. If asset and liability present values and Macaulay durations match at the same yield, their first-order changes are equal: the portfolio’s estimated surplus change is zero. This derivative argument assumes fixed cash flows and a common yield shift; it does not imply that actual changes are exactly equal for a finite shift.

Worked Example: Matching Two Zero-Coupon Bonds to a Liability

A hypothetical liability has present value $900 and Macaulay duration 3.2 years at an annual effective yield of 4%. A portfolio may use zero-coupon bonds paying at years 2 and 5. Find the present value assigned to each bond, their face amounts, and the first-order estimated surplus change for a 20-basis-point yield increase or decrease.

Method. A zero-coupon bond’s Macaulay duration is its payment time, so the asset durations are 2 and 5 years. Let $V_2$ and $V_5$ be the present values assigned to those bonds. The matching equations are

V_2+V_5=900,\qquad 2V_2+5V_5=900(3.2)=2880.

Subtract twice the first equation from the second: $3V_5=1080$, so $V_5=360$. Then $V_2=900-360=540$. The duration check is $(2(540)+5(360))/900=2880/900=3.2$ years.

For a zero-coupon bond, face amount equals present value multiplied by $(1.04)^n$. At year 2, $(1.04)^2=1.0816$, so the face amount is $540(1.0816)=\$584.064$, or $584.06 to the nearest cent. At year 5, $(1.04)^5=1.2166529024$, so the face amount is $360(1.2166529024)=\$437.995044864$, or $438.00 to the nearest cent.

The liability’s modified duration is $3.2/1.04=3.0769231$. For a 20-basis-point change, $|\Delta i|=0.002$, so the estimated absolute value change is $900(3.0769231)(0.002)=\$5.53846$, or $5.54 to the nearest cent. On a yield increase, the liability’s value is estimated to fall by $5.54; on a yield decrease, it is estimated to rise by $5.54. The matched assets have the same first-order change in either case, so the estimated surplus change is $0.00. This is a first-order estimate, not an assertion that the actual changes are identical.

Check and trap avoided. The assigned present values sum to $900 and reproduce duration 3.2. Do not confuse present value with face amount: the year-2 face amount is $540(1.04)^2=\$584.06$, not an amount obtained using the year-5 accumulation factor.

Worked Example: Matching a Coupon Bond and a Four-Year Zero

At an annual effective yield of 10%, a liability has present value $700 and Macaulay duration $239/77$ years. The available assets are a two-year bond with face amount $100, annual coupons of $10, and a $100 maturity payment, and a four-year zero-coupon bond with face amount $100. Find the market-value allocation to each asset, the number of two-year bonds, and the face amount of the four-year zeros. Then estimate the surplus change for a 10-basis-point yield increase.

Value and duration of the two-year bond. Its year-1 cash flow has present value $10/1.1=\$9.090909$. Its year-2 cash flow is $110, with present value $110/(1.1)^2=\$90.909091$. The price is therefore $100. Its duration is

D_1=\frac{1(9.090909)+2(90.909091)}{100} =\frac{21}{11}\text{ years}.

The four-year zero has duration $D_2=4$ years. Let $V_1$ be the market value assigned to the two-year bonds and $V_2$ the value assigned to the zeros. The target duration is $239/77$, so

V_1+V_2=700,\qquad \frac{21}{11}V_1+4V_2=700\frac{239}{77}.

Substitute $V_2=700-V_1$. The right side of the duration equation is $700(239/77)=2172.727273$. Therefore $4(700)-(4-21/11)V_1=2172.727273$, or $(23/11)V_1=700$. Hence $V_1=300$ and $V_2=400$. The portfolio duration check is $(300(21/11)+400(4))/700=239/77$.

Each two-year bond costs $100, so the required holding is $300/100=3$ bonds. A four-year zero with face amount $100 costs $100/(1.1)^4=\$68.30135$. The face amount needed to provide market value $400 is $400(1.1)^4=400(1.4641)=\$585.64$.

For a 10-basis-point increase, $\Delta i=0.001$. The matched liability’s first-order value change is $ -700(239/77)(0.001)/1.1=-\$1.97521$, or a decrease of $1.98 to the nearest cent. The assets have the same first-order dollar sensitivity because their total present value and duration match the liability. Their estimated change is also a decrease of $1.98, leaving an estimated surplus change of $0.00.

Check and trap avoided. The calculation uses the market value of each asset in the matching equations. The four-year zero’s $400 allocation is not its face amount; its face amount is $585.64. Also, the coupon bond’s duration is not simply its two-year maturity because some value is received at year 1.

Worked Example: A Small Upward and Downward Yield Shift

A liability has present value $1,200 and duration 3.5 years at a 6% annual effective yield. Match it with zero-coupon bonds paying at years 1 and 5. Find the market values and face amounts, then estimate the liability and asset value changes for a 10-basis-point shift in either direction.

Matching. Let $V_1$ and $V_5$ denote the two market values. The equations are $V_1+V_5=1200$ and $V_1+5V_5=1200(3.5)=4200$. Subtracting the first equation from the second gives $4V_5=3000$, so $V_5=\$750$ and $V_1=\$450$. The duration check is $(450+5(750))/1200=4200/1200=3.5$ years.

The year-1 face amount is $450(1.06)=\$477.00$. Since $(1.06)^5=1.3382255776$, the year-5 face amount is $750(1.3382255776)=\$1,003.6691832$, or $1,003.67 to the nearest cent. These face amounts correspond to the unrounded matched market values.

Modified duration is $3.5/1.06=3.3018868$. With $|\Delta i|=0.001$, the estimated absolute change for either the liability or the matched asset portfolio is $1200(3.3018868)(0.001)=\$3.96226$, or $3.96 to the nearest cent. For an upward shift, each is estimated to decrease by $3.96; for a downward shift, each is estimated to increase by $3.96. In both cases, the estimated surplus change is zero.

Check and trap avoided. The two assigned present values add to $1,200 and their weighted duration is 3.5 years. A yield decrease raises present values, so using the same negative sign for both upward and downward changes would be incorrect. The calculation uses the signed $\Delta i$ in $\Delta P\approx-PD\Delta i/(1+i)$.

Interpreting the Result and Avoiding Common Errors

A duration match is a local first-order hedge. The matched asset and liability values have equal estimated sensitivity at the valuation yield, so their estimated surplus change is zero for a sufficiently small parallel shift. For a larger shift, or when the yield curve changes nonparallelly, the actual asset and liability changes need not cancel. The first-order estimate also omits higher-order effects; it should not be treated as an exact repricing.

Using face amounts as weights. The duration equations use present values, not face amounts. Convert an allocation to face amount only after the match is solved.
Using maturity as every bond’s duration. This is true for a zero-coupon bond with one payment, but not generally for a coupon bond with earlier payments.
Forgetting the yield conversion. With annual effective yield $i$, the first-order change uses modified duration $D/(1+i)$, not Macaulay duration $D$ alone.
Claiming exact protection. Matching PV and duration cancels the first-order estimate under the stated assumptions; it does not prove exact equality after a nonzero yield shift.
Ignoring feasibility. If the liability duration is outside the range of the two asset durations, at least one solved market value is negative. A positive-only portfolio cannot achieve that match with those two assets.

Key takeaway: First solve the market-value equations $V_1+V_2=L$ and $V_1D_1+V_2D_2=LD_L$. Then translate the allocations into actual holdings and check both conditions. Under a small parallel yield shift, use modified duration to estimate changes; a match makes the asset and liability estimates equal to first order.

Check Your Understanding

For each problem, assume fixed cash flows, annual effective yield, and a small parallel yield shift where one is requested. Round money to the nearest cent.

At 5%, a liability has present value $500 and duration 4 years. Match it with zero-coupon bonds paying at years 2 and 6. Give the market value assigned to each and the face amounts. (Answer: $250 at each maturity; face amounts $275.63 at year 2 and $335.02 at year 6.)
At an 8% annual effective yield, a liability has present value $1,000 and duration 3 years. Match it with zero-coupon bonds paying at years 1 and 5. Give the market values and face amounts. (Answer: $500 at each maturity; face amounts $540.00 at year 1 and $734.66 at year 5.)
For the matched portfolio in question 2, estimate the liability’s value change for a 25-basis-point yield increase. What is the first-order estimated surplus change? (Answer: Liability decrease of $6.94; estimated surplus change $0.00.)
A liability has present value $900 and duration 3 years. Two assets have durations 1.5 and 4 years. Find the market value assigned to each asset. (Answer: $540 at duration 1.5 years and $360 at duration 4 years.)

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