Positive-Weight Feasibility from Duration Bracketing

Why Duration Bracketing Matters

Matching a liability’s present value sets the portfolio’s value at the chosen valuation rate. Matching duration aligns the portfolio’s first-order response to a small change in that rate. But can those two conditions be met using only positive investments in the available assets? Duration bracketing answers that feasibility question before you solve for holdings.

The key idea is that a portfolio made from positive positions has a duration that is a present-value-weighted average of the durations of its assets. Its duration therefore cannot fall below the smallest asset duration or rise above the largest. If the liability duration lies strictly between two asset durations, positive allocations to those two assets can match both present value and duration. That is a feasibility result; it does not establish that the portfolio also passes the separate convexity test.

Definition: At a valuation rate $i$, the present value of deterministic cash flows $C_t$ paid at times $t$ is $P(i)=\sum_t C_t(1+i)^{-t}$. Their Macaulay duration is $D(i)=\sum_t tC_t(1+i)^{-t}/P(i)$. For a portfolio of assets with positive holdings, its duration is the total present-value-weighted average of the asset durations. In this tutorial, a positive allocation means a strictly positive amount invested in an asset at the valuation rate.

Use a consistent valuation rate and time unit throughout: if cash-flow times are measured in years, durations are in years. Assume the asset and liability cash flows are deterministic and have positive present values. For a portfolio with holdings $H_j$, asset prices $P_j$, and durations $D_j$, define each present-value allocation as $x_j=H_jP_j$. Its present value is $\sum_j x_j$, and its duration is $\sum_jx_jD_j/\sum_jx_j$.

The Bracketing Test and the Two-Asset Solution

Suppose a liability has present value $L_0$ and duration $D_L$, and two assets have positive prices and durations $D_1<D_2$. Let $x_1$ and $x_2$ be the present-value amounts invested in the assets. Matching present value and duration requires two equations:

x_1+x_2=L_0,\qquad x_1D_1+x_2D_2=L_0D_L.

Solving the equations gives the present-value allocations directly. Holdings in units of each asset are then found by dividing the allocation by that asset’s price.

x_1=L_0\frac{D_2-D_L}{D_2-D_1},\qquad x_2=L_0\frac{D_L-D_1}{D_2-D_1}.

When $L_0>0$ and $D_1<D_2$, both allocations are strictly positive exactly when $D_1<D_L<D_2$. If $D_L=D_1$, the solution puts the entire present value in asset 1 and assigns zero to asset 2; if $D_L=D_2$, it does the reverse. If the liability duration is outside the closed interval from $D_1$ to $D_2$, the equations require a negative allocation to one asset.

Theorem: A positive-weight portfolio of two assets with durations $D_1<D_2$ can match a liability’s present value and duration with both weights strictly positive if and only if $D_1<D_L<D_2$. For a selection from several positive-price assets, a liability duration strictly between the minimum and maximum asset durations can be matched using positive allocations in a suitable pair that brackets it. This result concerns present value and duration only.

Check the range.
Compare the liability duration with the smallest and largest available asset durations. If it is outside their range, positive matching allocations are impossible.

Choose a bracketing pair.
For a two-asset construction, select assets whose durations lie on opposite sides of the liability duration.

Solve for present-value allocations.
Use the two matching equations, then convert each allocation into holdings using the corresponding asset price.

Test convexity separately.
After present value and duration match, compare the asset and liability second derivatives at the valuation rate.

Why Duration Is a Weighted Average

For asset $j$, let $P_j$ be its price and $D_j$ its duration. The portfolio price is $\sum_j H_jP_j$. Its duration numerator is $\sum_j H_jP_jD_j$, because every cash flow of asset $j$ is multiplied by its holding $H_j$. Consequently,

D_{\text{portfolio}} =\frac{\sum_jH_jP_jD_j}{\sum_jH_jP_j} =\sum_j\left(\frac{H_jP_j}{\sum_kH_kP_k}\right)D_j.

With positive holdings and prices, each coefficient in parentheses is positive, and the coefficients sum to 1. Thus the portfolio duration is a weighted average of the asset durations. For two assets, setting that weighted average equal to $D_L$, while setting total present value equal to $L_0$, yields the allocation formulas above. This also explains the feasibility test: a positive weighted average of two distinct numbers lies strictly between them.

Worked Example: Constructing a Positive Two-Asset Match

At an effective annual valuation rate of 4%, a liability is a deterministic payment of $1,459.98 at time 5. Its present value is $1,200. Two available assets are zero-coupon bonds: asset A pays $103.83 at time 2 and has a price of $96; asset B pays $142.33 at time 8 and has a price of $104. Use the stated prices and liability present value to construct a positive present-value-and-duration match. There are no mortality-contingent payments.

Method: duration bracketing and present-value allocation. A zero-coupon bond’s duration is its payment time, so $D_A=2$, $D_B=8$, and the liability duration is $D_L=5$. The liability duration lies strictly between the asset durations, so positive allocations are feasible.

Let $x_A$ and $x_B$ be the dollars of present value allocated to the assets. Matching present value gives $x_A+x_B=1{,}200$. Matching duration gives $2x_A+8x_B=1{,}200(5)=6{,}000$. Substituting $x_A=1{,}200-x_B$ into the duration equation gives $2(1{,}200-x_B)+8x_B=6{,}000$, so $2{,}400+6x_B=6{,}000$, $x_B=600$, and $x_A=600$.

Convert these allocations into units: holdings of A are $600/96=6.25$ bonds, and holdings of B are $600/104=75/13$, or approximately 5.7692 bonds. Both holdings are positive.

Check. The portfolio present value is $6.25(96)+(75/13)(104)=600+600=$1{,}200$. Its duration is $(600(2)+600(8))/1{,}200=5$, matching the liability. The common mistake is to calculate weights from the bond prices alone; the matching equations use each asset’s present-value contribution.

Matching Is Not the Same as Immunizing

Duration bracketing proves that positive weights can meet the present-value and duration equations. Redington immunization also requires the matched surplus to have positive second derivative at the valuation rate. To state that test precisely, let $A(i)$ and $L(i)$ be asset and liability present values, and define the surplus $S(i)=A(i)-L(i)$. For deterministic cash flows,

P'(i)=-\frac{P(i)D(i)}{1+i},\qquad P''(i)=\sum_t t(t+1)C_t(1+i)^{-t-2}.

If asset and liability present values and durations match at $i_0$, then their first derivatives match there as well, so $S(i_0)=0$ and $S'(i_0)=0$. The remaining local Redington condition is $S''(i_0)=A''(i_0)-L''(i_0)>0$. Equivalently, define the second-derivative convexity $K=P''/P$; since the matched present values are equal, the condition is $K_A>K_L$. This is a local condition at the chosen rate, not a guarantee about every possible rate change.

Important distinction: Duration bracketing addresses whether positive present-value and duration matching weights exist. It does not determine whether the resulting asset portfolio has greater second-derivative convexity than the liability. Compute the convexity comparison from the cash flows after matching.

Worked Example: Bracketing Followed by a Successful Convexity Test

At an effective annual valuation rate of 5%, a liability is a deterministic zero-coupon payment at time 5 with present value $1,000. Two zero-coupon assets pay at times 2 and 8. Choose positions that match the liability’s present value and duration, then test the second-derivative condition. There are no mortality-contingent payments.

Method: match the duration using present-value allocations, then compare second derivatives. The asset durations are 2 and 8, which bracket the liability duration of 5. The allocation formulas give $x_2=x_8=1{,}000/2=$500$. The corresponding payments are $500(1.05)^2=$551.25$ at time 2 and $500(1.05)^8=$738.73$ at time 8, rounded to cents. The liability payment is $1{,}000(1.05)^5=$1{,}276.28$, rounded to cents.

The asset duration is $(500(2)+500(8))/1{,}000=5$, and the liability duration is 5 because it pays at time 5. Both present values are $1,000. For a cash-flow stream, $P''$ at the valuation rate is the sum of each cash flow’s present value multiplied by $t(t+1)/(1+i)^2$. Therefore the asset second derivative is $(500(2)(3)+500(8)(9))/(1.05)^2=39{,}000/1.1025$. The liability second derivative is $1{,}000(5)(6)/(1.05)^2=30{,}000/1.1025$.

The surplus second derivative is $(39{,}000-30{,}000)/1.1025=$8{,}163.27$ per squared unit of the annual rate, rounded to cents. It is positive, so this portfolio passes the local second-derivative test. The common mistake is to stop when the durations match; here the separate calculation shows why the convexity condition also holds.

Worked Example: A Duration Match That Fails the Convexity Test

At an effective annual valuation rate of 5%, a hypothetical asset portfolio has two deterministic payments, each with present value $500, at times 4 and 6. A liability has two deterministic payments, each with present value $500, at times 1 and 9. Assess present-value matching, duration matching, and the local second-derivative condition. There are no mortality-contingent payments.

Method: compare both duration and second-derivative convexity. The asset present value is $500+500=$1{,}000$, and its duration is $(500(4)+500(6))/1{,}000=5$. The liability present value is also $1,000, and its duration is $(500(1)+500(9))/1{,}000=5$. Thus present value and duration match. The corresponding payments are determined by multiplying each stated present value by $1.05^t$; no further payment amounts are needed for the present-value-weighted calculations.

The asset second derivative is $[500(4)(5)+500(6)(7)]/(1.05)^2=31{,}000/1.1025$. The liability second derivative is $[500(1)(2)+500(9)(10)]/(1.05)^2=46{,}000/1.1025$. Hence the surplus second derivative is $(31{,}000-46{,}000)/1.1025=-$13{,}605.44$ per squared unit of the annual rate, rounded to cents.

The second derivative is negative, so the portfolio fails the local Redington convexity condition despite matching present value and duration. The common mistake is to assume that bracketing and matching durations automatically imply immunization. They establish feasibility for the first two conditions only.

Common Traps in Feasibility Problems

Using an ordinary average instead of a present-value-weighted average. The weights are each asset’s share of portfolio present value, not an unadjusted average of the asset durations.
Calling a boundary solution strictly positive. If the liability duration equals one asset’s duration, matching may be possible, but the allocation to a different-duration asset must be zero.
Ignoring the direction of the bracket. A liability duration above every available asset duration cannot be reached with positive holdings; solving the equations will expose this as a negative allocation.
Treating a duration match as a complete Redington test. Check $A''(i_0)-L''(i_0)>0$ separately after present value and duration have been matched.

Check Your Understanding

For each problem, assume deterministic cash flows, positive asset prices, and the stated common valuation rate. Decide whether the requested positive match is feasible and calculate any requested amounts.

A liability has present value $750 and duration 6. Two assets have durations 1 and 9. Find the present-value allocation to each asset. (Answer: $281.25 to the duration-1 asset and $468.75 to the duration-9 asset; both are positive.)
A liability has present value $500 and duration 4. The only two assets have durations 4 and 9. Can both assets receive strictly positive allocations while matching present value and duration? (Answer: No; the duration-4 asset must receive all $500, and the duration-9 allocation is zero.)
A liability has present value $1,200 and duration 6. Two available assets have durations 2 and 8. Can a positive two-asset present-value-and-duration match be constructed? If so, give the present-value allocations. (Answer: Yes; $400 to the duration-2 asset and $800 to the duration-8 asset.)
At a common valuation rate of 5%, an asset portfolio has present-value cash-flow amounts of $400 at time 2 and $600 at time 8. A liability has present value $1,000 and duration 5. Find the asset duration. Does the asset portfolio match duration? (Answer: Asset duration is 5.6; it does not match the liability duration.)
At a common valuation rate of 5%, asset present-value cash flows are $500 at time 3 and $500 at time 7; liability present-value cash flows are $500 at time 1 and $500 at time 9. Compare durations and determine whether the surplus second derivative is positive. (Answer: Both durations are 5 and both present values are $1,000, but the surplus second derivative is $[500(12+56-2-90)]/1.1025=-$10{,}884.35$, so the convexity condition fails.)

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