Nonnegative Holdings in Duration Matching

Why Long-Only Feasibility Matters

A duration equation can have an algebraic solution that is not a usable portfolio. In particular, solving for weights may assign a negative amount to one asset. That result can describe a short position, but it does not describe a long-only portfolio. Before calculating a proposed allocation, test whether the liability’s duration lies within the range the available assets can produce.

This tutorial considers fixed cash flows valued at a common effective annual yield, with duration measured consistently for the assets and liability. The focus is on whether nonnegative market-value holdings can attain a target duration—not on whether those holdings reproduce every liability payment.

Definition: Let $D_j$ be the duration of available asset $j$, and let $x_j$ be the market value invested in it. If total asset value is $V=\sum_j x_j>0$, the portfolio duration is $$ D_P=\sum_j w_jD_j,\qquad w_j=\frac{x_j}{V}. $$ A long-only portfolio has $x_j\geq 0$ for every asset, so its value weights satisfy $w_j\geq0$ and $\sum_jw_j=1$. A duration match also requires the portfolio value to equal the liability value; the duration equation alone does not set the required funding amount.

For fixed cash flows discounted at a common effective annual yield, Macaulay duration is the present-value-weighted average payment time. If a cash flow $C_t$ is paid at time $t$, its present value is $C_tv^t$, where $v=1/(1+i)$. Thus the duration is the sum of each payment time multiplied by its share of present value. The portfolio formula above follows by applying this same weighting idea first within each asset and then across asset values.

The Feasible Duration Range

Let $D_{\min}$ and $D_{\max}$ be the smallest and largest durations among the assets that may be held. A long-only portfolio cannot have a duration below $D_{\min}$ or above $D_{\max}$: it is an average of the available durations, with nonnegative weights summing to one.

Theorem: With freely divisible holdings, a target duration $D_L$ is attainable by a long-only portfolio if and only if $$ D_{\min}\leq D_L\leq D_{\max}. $$ If $D_L$ is strictly between the endpoints, a mix of an asset at $D_{\min}$ and an asset at $D_{\max}$ attains it. If $D_L=D_{\min}$ or $D_L=D_{\max}$, only assets with that endpoint duration can receive positive weight in an exact match.

The phrase “freely divisible” matters. If a separate rule limits the amount invested in an asset, requires whole units, or prohibits a particular holding, being inside the duration range may not be sufficient. The result here assumes that any nonnegative value allocation among the stated assets is allowed.

Why the Range Test Works

The range test is a direct consequence of weighted averages. Since $D_{\min}\leq D_j\leq D_{\max}$, multiplying each inequality by $w_j\geq0$ and summing gives $\sum_jw_jD_{\min}\leq\sum_jw_jD_j\leq\sum_jw_jD_{\max}$. Because the weights sum to one, this becomes $D_{\min}\leq D_P\leq D_{\max}$.

For sufficiency, choose one asset with duration $D_{\min}$ and one with duration $D_{\max}$. If $D_{\min}<D_L<D_{\max}$, assign fraction $\alpha$ to the maximum-duration asset and $1-\alpha$ to the minimum-duration asset. Solving the duration equation gives

D_L=(1-\alpha)D_{\min}+\alpha D_{\max}, \qquad \alpha=\frac{D_L-D_{\min}}{D_{\max}-D_{\min}}.

The target being between the endpoints ensures $0<\alpha<1$, so neither fraction is negative. At an endpoint, the formula gives a fraction of zero or one. This construction proves that every target in the interval is attainable under the stated assumptions.

Check values.
Find the liability present value and the total amount available for investment. A duration match must also fund the liability at the valuation date.

Find the range.
Identify the minimum and maximum durations among the eligible assets.

Test the target.
If the liability duration is outside the range, no long-only duration match exists. If it is inside, construct nonnegative weights and verify their sum and weighted duration.

Convert weights to values.
Multiply each weight by the required total portfolio value. Do not confuse a value weight with a fraction of face amount or number of securities.

Worked Allocations and Infeasible Targets

Worked Example: Matching a Liability Between Two Available Durations

A hypothetical liability has present value $ \$800{,}000 $ and duration 5 years. Three eligible fixed-cash-flow assets have durations 2, 5, and 8 years. Determine whether a long-only match is possible and give one allocation.

Method. First compare the target with the duration range, then solve for a value-weighted combination. This avoids assuming that the asset with the closest duration must be held by itself.

The available range is from 2 to 8 years, and $2\leq5\leq8$, so a long-only match is feasible. Use the 2-year and 8-year assets. Let $\alpha$ be the fraction invested in the 8-year asset:

5=(1-\alpha)2+\alpha 8 =2+6\alpha, \qquad \alpha=\frac{3}{6}=0.5.

The remaining fraction is $1-0.5=0.5$. The market values are $0.5(\$800{,}000)=\$400{,}000$ in each asset. The 5-year asset is not needed in this particular allocation.

Check. The values total $\$400{,}000+\$400{,}000=\$800{,}000$, and $0.5(2)+0.5(8)=1+4=5$ years. Both holdings are nonnegative, so this is a valid long-only duration match.

Worked Example: A Target at the Upper Endpoint

A hypothetical liability has present value $\$1{,}200{,}000$ and duration 7 years. Assets with durations 3, 5, and 7 years are available. Find a long-only match if one exists.

The range is $[3,7]$, so the target equals the upper endpoint and is feasible. To average to the maximum available duration of 7, every asset receiving positive weight must itself have duration 7. Put the full $\$1{,}200{,}000$ into the 7-year asset and zero into the other two.

Check. The values sum to $\$1{,}200{,}000$, and the portfolio duration is $0(3)+0(5)+1(7)=7$ years. A positive holding in either shorter-duration asset would pull the weighted average below 7.

Worked Example: Diagnosing a Negative-Holding Proposal

A hypothetical liability has duration 6.4 years. Available assets have durations 1, 4, and 6 years. An analyst proposes investing 110% of portfolio value in the 6-year asset and negative 10% in the 1-year asset. Is the proposal a match, and can any long-only match reach the target?

Method. Check the stated weighted duration and the feasibility range separately. A negative weight fails the long-only condition even if the duration equation happens to be correct.

The proposed duration is

1.10(6)+(-0.10)(1)=6.60-0.10=6.50\text{ years}.

Therefore, the proposed weights do not even match the stated 6.4-year target. They also include a negative holding, so they are not long-only.

To see what an algebraic match using only the 1-year and 6-year assets would require, let $w_6$ be the fraction in the 6-year asset:

6.4=w_6(6)+(1-w_6)(1) =1+5w_6, \qquad w_6=\frac{5.4}{5}=1.08.

The 1-year fraction would be $1-1.08=-0.08$. This gives an algebraic match, since $1.08(6)+(-0.08)(1)=6.48-0.08=6.40$, but it requires a negative holding. More decisively, the maximum available duration is 6 years and $6.4>6$, so no long-only match exists.

Worked Example: Converting a Feasible Weight to Dollar Holdings

A hypothetical liability has present value $\$600{,}000$ and duration 4.5 years. The eligible assets have durations 2, 4, and 7 years. Construct a long-only portfolio using only the 2-year and 7-year assets, and verify it.

The target is within the range $[2,7]$. Let $\alpha$ be the value fraction in the 7-year asset. Then

4.5=(1-\alpha)2+\alpha7 =2+5\alpha, \qquad \alpha=\frac{2.5}{5}=0.5.

Thus the fraction in the 2-year asset is also $0.5$. The dollar holdings are $0.5(\$600{,}000)=\$300{,}000$ in each asset.

Check. The holdings total $\$600{,}000$, and the duration is $0.5(2)+0.5(7)=1+3.5=4.5$ years. Both fractions are nonnegative, so the allocation is long-only.

Interpretation and Common Traps

A target inside the range is a necessary and sufficient feasibility condition only for the duration equation under the assumptions stated above. It does not show that the portfolio has the same dated cash flows as the liability. Nor does a duration match guarantee that values move identically for every possible change in interest rates. Duration is a summary of cash-flow timing and, in matching applications, a first-order sensitivity measure.

Do not use face amounts as weights. The weights in the portfolio-duration average are proportions of market value, not proportions of par value or units held.
Do not accept a negative fraction as long-only. A negative weight represents borrowing or a short position. It may solve an unconstrained equation, but it violates the long-only requirement.
Do not skip the range test. If $D_L>D_{\max}$ or $D_L<D_{\min}$, changing the mix among the same assets cannot fix the problem.
Do not infer cash-flow matching. Equal present value and duration are only two aggregate conditions; they do not require payment-by-payment equality.
Use consistent duration definitions and valuation assumptions. Comparing durations calculated on incompatible yield bases does not support the weighted-average feasibility test.

This feasibility check is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In an exam calculation, state the duration range, compare the target with both endpoints, and then verify the proposed weights explicitly.

Check Your Understanding

For each problem, test feasibility before calculating an allocation. All holdings are freely divisible, and the listed durations are measured consistently.

A liability has present value $\$450{,}000$ and duration 4 years. Available asset durations are 1, 4, and 6 years. Give a long-only allocation using the 1-year and 6-year assets, in dollars. (Answer: $\$180{,}000$ in the 1-year asset and $\$270{,}000$ in the 6-year asset; the weights are 0.4 and 0.6, giving duration $0.4(1)+0.6(6)=4$.)
A liability has duration 2.5 years. The available durations are 3, 5, and 8 years. Is a long-only match possible? (Answer: No; 2.5 is below the minimum available duration of 3 years.)
A liability has duration 6 years, and the only eligible asset durations are 2 and 10 years. What fraction of value must be invested in the 10-year asset? (Answer: $0.5$; solving $6=2(1-\alpha)+10\alpha$ gives $\alpha=0.5$, with the other 0.5 in the 2-year asset.)
A liability has present value $\$900{,}000$ and duration 7 years. Available assets have durations 4, 7, and 9 years. Give one long-only allocation that matches both value and duration. (Answer: Invest $\$900{,}000$ in the 7-year asset and zero in the others; its duration is 7 years.)
A liability has duration 5.5 years. The available assets have durations 2, 4, and 5 years. An analyst proposes weights 1.1 in the 5-year asset and $-0.1$ in the 2-year asset. Does this produce a long-only match, and what is the feasibility conclusion? (Answer: No. Its duration is $1.1(5)-0.1(2)=5.3$ years, not 5.5; moreover, the target exceeds the maximum available duration of 5 years, so no long-only match exists.)

What You'll Learn