Solving Asset Weights from Present Value and Duration

Why Solve for Asset Weights?

A liability may have a known present value and duration at a specified valuation rate. A portfolio of two assets can be chosen to match both quantities by solving two linear equations. The key is to keep each asset’s investment amount distinct from its number of units: duration equations are simplest when written in terms of the present-value contribution of each asset.

The calculation is a two-condition match, not a complete immunization test. Matching present value and duration sets the portfolio’s value and first-order rate sensitivity equal to those of the liability at the valuation rate. It does not, by itself, establish that the portfolio satisfies a convexity condition. The method here is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: At a common valuation rate, let $L$ be the present value of a liability and $D_L$ its Macaulay duration. For asset $j$, let $P_j$ be its price per unit and $n_j$ the number of units held. Its present-value contribution is $z_j=n_jP_j$, and its duration is $D_j$. A two-asset present-value and duration match satisfies $z_1+z_2=L$ and $z_1D_1+z_2D_2=LD_L$.

Set Up the Two Equations

Assume all values are measured at the same date and valuation rate, and each asset and liability is represented by fixed dated cash flows. For an effective annual valuation rate $i$, a cash flow $C_t$ payable at time $t$ has present value $C_tv^t$, where $v=1/(1+i)$. For cash flows $C_t$, Macaulay duration is the present-value-weighted average payment time:

D=\frac{\sum_t tC_tv^t}{\sum_t C_tv^t}.

The portfolio’s present value is the sum of its assets’ present values. Its duration is the average of their durations weighted by those present values. Therefore, the two conditions for matching a liability are:

z_1+z_2=L, \qquad z_1D_1+z_2D_2=LD_L.

The first equation matches value. The second matches the present-value-weighted time of the cash flows. It is a dollar-duration equation: it uses each asset’s present value multiplied by its duration, rather than simply adding the assets’ durations.

Deriving the Direct Weight Formulas

When the asset durations differ, $D_1\ne D_2$, the two equations have a unique algebraic solution. Substitute $z_2=L-z_1$ from the value equation into the duration equation:

$$ z_1D_1+(L-z_1)D_2=LD_L. $$

Collecting the terms involving $z_1$ gives $z_1(D_1-D_2)=L(D_L-D_2)$. Solving, and then using $z_2=L-z_1$, gives:

z_1=L\frac{D_2-D_L}{D_2-D_1}, \qquad z_2=L\frac{D_L-D_1}{D_2-D_1}.

These are present-value contributions, not necessarily numbers of units. To find the holdings, divide each contribution by the corresponding asset’s price per unit: $n_j=z_j/P_j$. This distinction matters especially when the assets have different prices.

There is also a useful rate-sensitivity justification for the second equation. For fixed cash flows, the price of an asset is $P(i)=\sum_t C_tv^t$. Differentiating with respect to the effective annual rate gives $dP/di=-\sum_t tC_tv^{t+1}$. Since $\sum_t tC_tv^t=PD$, it follows that:

\frac{dP}{di}=-\frac{PD}{1+i}.

Thus matching both present value and duration matches the portfolios’ first derivatives of value with respect to the valuation rate. This explains the first-order rate-sensitivity interpretation; it does not guarantee equality of values after a finite rate change.

Value the liability and assets.
Use one valuation date, one effective annual rate, and consistent cash-flow timing. Record each asset’s price per unit.

Calculate durations.
Use the present-value-weighted payment times for each asset and the liability.

Solve for present-value contributions.
Apply the two equations or the direct formulas for $z_1$ and $z_2$.

Convert to holdings and verify.
Set $n_j=z_j/P_j$, then substitute the contributions into both matching equations.

Worked Calculations

Worked Example: Two Zero-Coupon Assets at 10%

A liability pays $1,331 at the end of year 3. At an effective annual valuation rate of 10%, an investor can buy one-year and five-year zero-coupon assets, each paying $1 per unit at maturity. Find the number of units of each asset needed to match the liability’s present value and duration.

Method. First match the present-value contributions, then convert them to units. The liability’s present value is:

L=1{,}331(1.1)^{-3} =\frac{1{,}331}{1.331} =1{,}000.

A single payment at year 3 has duration $D_L=3$. The zero-coupon assets have durations $D_1=1$ and $D_2=5$, because each asset’s only payment is at its maturity. The contribution equations are:

z_1+z_2=1{,}000, \qquad z_1+5z_2=3(1{,}000)=3{,}000.

Subtract the first equation from the second: $4z_2=2{,}000$, so $z_2=500$. Then $z_1=1{,}000-500=500$. The present-value contributions are $500 in each asset.

The one-year asset costs $P_1=(1.1)^{-1}=1/1.1$ per unit. Its required holding is $n_1=500/(1/1.1)=550$ units. The five-year asset costs $P_2=(1.1)^{-5}=1/1.61051$ per unit, so $n_2=500/(1/1.61051)=805.255$ units.

Verification. The present values of the holdings are $550/1.1=500$ and $805.255/1.61051=500$. Their total is $1,000. Their dollar-duration is $500(1)+500(5)=3{,}000$, giving portfolio duration $3{,}000/1{,}000=3$ years, equal to the liability duration. The five-year unit count is shown to three decimal places; calculations before rounding use $500(1.1)^5=805.255$ exactly.

Common mistake avoided: The weights $500$ and $500$ are dollars of present value, not units. Using them as unit counts would not produce the required present values because the two assets have different prices.

Worked Example: A Single Liability Payment and Unequal Asset Prices

At an effective annual valuation rate of 5%, a liability payment of $121.550625 is due at the end of year 4. Two available assets are zero-coupon securities, one maturing at year 2 and one at year 6, each paying $1 per unit at maturity. Determine the holdings that match value and duration.

The liability present value is $121.550625/(1.05)^4=100$, since $(1.05)^4=1.21550625$. Its duration is $D_L=4$. The asset durations are $D_1=2$ and $D_2=6$. The equations for present-value contributions are:

z_1+z_2=100, \qquad 2z_1+6z_2=4(100)=400.

Subtract twice the first equation from the second: $4z_2=200$, so $z_2=50$, and therefore $z_1=50$. Each asset must contribute $50 of present value.

The unit prices are $P_1=(1.05)^{-2}$ and $P_2=(1.05)^{-6}$. Thus:

n_1=50(1.05)^2=55.125, \qquad n_2=50(1.05)^6=67.00478203125.

Verification. Discounting the holdings’ maturity payments at 5% gives a present value of $50 for each asset, hence $100 in total. The portfolio’s dollar-duration is $50(2)+50(6)=400$, so its duration is $400/100=4$ years. The second holding is 67.005 units when rounded to three decimal places; the unrounded value is used for verification.

Common mistake avoided: The liability’s payment amount, $121.550625, is not its present value. The matching equations use $100, the payment discounted four years at the specified valuation rate.

Worked Example: Matching with Coupon Assets

At an effective annual valuation rate of 10%, a liability pays $121 at year 2 and $146.41 at year 4. Asset A pays $44 at year 1 and $53.24 at year 3 per unit. Asset B pays $24.20 at year 2 and $87.8464 at year 4 per unit. Find the number of units of each asset that match the liability’s present value and duration.

Method. Calculate each cash flow’s present value first, then use those present values to calculate duration. The liability cash flows have present values $121/(1.1)^2=100$ at year 2 and $146.41/(1.1)^4=100$ at year 4. Therefore $L=200$, and:

D_L=\frac{2(100)+4(100)}{200}=3.

For Asset A, the year-1 cash flow has present value $44/1.1=40$, and the year-3 cash flow has present value $53.24/(1.1)^3=40$. Its price is $P_A=40+40=80$, and its duration is $(1(40)+3(40))/80=2$ years.

For Asset B, the year-2 cash flow has present value $24.20/(1.1)^2=20$, and the year-4 cash flow has present value $87.8464/(1.1)^4=60$. Its price is $P_B=20+60=80$, and its duration is:

D_B=\frac{2(20)+4(60)}{80} =\frac{280}{80} =3.5 \text{ years}.

Let $z_A$ and $z_B$ be the assets’ present-value contributions. Matching value and duration requires:

z_A+z_B=200, \qquad 2z_A+3.5z_B=3(200)=600.

Subtract $2(z_A+z_B)=400$ from the second equation: $1.5z_B=200$, so $z_B=133.333\ldots$ and $z_A=66.666\ldots$. Since both assets cost $80 per unit, the required holdings are $n_A=z_A/80=5/6$ unit and $n_B=z_B/80=5/3$ units.

Verification. The present values are $(5/6)(80)=66.666\ldots$ and $(5/3)(80)=133.333\ldots$, totaling $200. The dollar-duration is $2(66.666\ldots)+3.5(133.333\ldots)=600$, so portfolio duration is $600/200=3$ years. Holdings are exact fractions; displayed decimals are rounded only for explanation.

Common mistake avoided: Asset B’s duration is not the average of years 2 and 4. Its year-4 cash flow has three times the present value of its year-2 cash flow, so the duration must be weighted accordingly.

When the Solution Is Unique—and What It Does Not Establish

If $D_1\ne D_2$, the two linear equations have a unique solution for the present-value contributions. If $D_1=D_2$, every portfolio made from the two assets has that same duration whenever its total present value is fixed. In that case, the equations can be satisfied only if $D_L=D_1=D_2$; if so, they determine the total contribution but not how it is divided between the assets.

After solving, check both equations using the unrounded contributions. If holdings are required, also divide by the correct per-unit prices. A match of value and duration alone does not check a separate requirement on asset cash flows, restrictions on holdings, or convexity. In particular, do not infer a strict convexity condition merely from a successful two-equation match.

Key takeaway: Solve for present-value contributions first: their sum must equal the liability present value, and their duration-weighted sum must equal liability present value times liability duration. Convert contributions to asset units only after solving, and verify both conditions.

Check Your Understanding

For each problem, use the stated valuation rate and treat the given durations as Macaulay durations at that rate.

At 8%, a liability has present value $240 and duration 4 years. Two zero-coupon assets mature at years 1 and 7 and each pays $1 per unit. Find their required present-value contributions and unit holdings. (Answer: Contributions $120 each; holdings 129.600 and 205.659 units, respectively.)
Two assets have durations 3 and 5 years. A liability has present value $900 and duration 4.5 years. Find each asset’s present-value contribution. (Answer: $225 to the duration-3 asset and $675 to the duration-5 asset.)
Two assets have durations 1 and 4 years. A liability has present value $500 and duration 2.2 years. Find the contributions and verify the dollar-duration. (Answer: $300 and $200, respectively; dollar-duration $300(1)+200(4)=1{,}100$, giving duration 2.2 years.)
At 10%, Asset A pays $22 at year 1 and $26.62 at year 3 per unit; Asset B pays $12.10 at year 2 and $43.9232 at year 4 per unit. A liability has present value $120 and duration 3 years. Find the asset holdings. (Answer: Each asset costs $40 and has duration 2 and 3.5 years, respectively; invest $60 present value in each, so hold 1.5 units of each.)
A liability has present value $400 and duration 5 years. Two assets have durations 2 and 6 years and per-unit prices of $96 and $80, respectively. Find the holdings that match value and duration. (Answer: Present-value contributions $100 and $300; holdings $100/96=1.0417$ and $300/80=3.75$ units.)

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