From Matching Targets to a Fundable Portfolio
A practical asset-liability problem rarely ends when the asset portfolio has the same present value and duration as the liabilities. Holdings may be restricted, cash may arrive on the wrong dates, and a nonparallel change in rates can affect assets and liabilities differently. A complete selection therefore checks both the matching equations and the portfolio’s ability to meet dated payments under the stated reinvestment and stress assumptions.
The examples use deterministic cash flows, no default, no taxes or transaction costs, and a common base valuation basis wherever present values and durations are compared. A holding is measured in units of an asset. Stressed market values are scenario inputs; they are not inferred from a duration approximation unless stated.
The dollar-duration equation matches the present-value-weighted timing of the asset cash flows to that of the liabilities. It does not, by itself, require assets and liabilities to have equal cash flows on every date. Nonnegative holdings and any liquidity, maturity, or concentration limits must be checked separately.
Why the Duration Equation Uses Dollar Duration
For deterministic cash flows \(C_k\) paid at times \(t_k\), discounted on the same annual effective yield basis, present value is the sum of discounted payments. Macaulay duration is the average payment time, weighted by each payment’s share of present value. The portfolio formula follows directly from those weights.
To see why, write the present value of asset \(j\) as the sum of the present values of its individual payments. Its duration numerator is the sum of each discounted payment multiplied by its payment time. Adding those numerators across assets gives \(\sum_jx_jP_jD_j\). Dividing by total asset present value gives the portfolio’s weighted-average payment time. Thus matching \(P_A=P_L\) and \(D_A=D_L\) is equivalent to matching both present value and the dollar-duration amount \(P_AD_A=P_LD_L\). The result assumes a common valuation basis and consistent duration definitions; it is not a statement that the cash flows match date by date.
Worked Example: Solving With a Fixed Holding and Nonnegativity
At a 5% annual effective valuation yield, three zero-coupon assets each have a current price of $1,000 per unit and mature in 1, 2, and 3 years. A deterministic liability portfolio has present value $1,500 and Macaulay duration \(7/3\) years on the same basis. Asset 3 must be held at 0.5 units. Find holdings \(x_1\) and \(x_2\) that match present value and duration, and check feasibility.
Method. Match present value first, then match the dollar-duration amount. Since the zero-coupon assets mature in 1, 2, and 3 years, their durations are 1, 2, and 3 years, respectively.
The liability dollar-duration amount is
The present-value equation, in dollars, is
so \(x_1+x_2=1\). The duration equation is
so \(x_1+2x_2=2\). Subtracting \(x_1+x_2=1\) from this equation gives \(x_2=1\), and then \(x_1=0\). Therefore the holdings are \(x_1=0\), \(x_2=1\), and \(x_3=0.5\).
Check. Present value is \(0(1{,}000)+1(1{,}000)+0.5(1{,}000)=1{,}500\). Dollar duration is \(0(1{,}000)(1)+1(1{,}000)(2)+0.5(1{,}000)(3)=3{,}500\), matching the liability amount. All holdings are nonnegative, so the solution is feasible. The common mistake is to solve only the present-value equation: that equation alone does not determine the two free holdings or establish duration matching.
Dated Payments and the Reinvestment Plan
A present-value match is measured at one valuation date. Funding a liability also means having enough cash available when each payment falls due. If an asset pays before a liability, the plan must specify what happens to the interim cash. For example, it may be reinvested at an assumed rate. That assumption is part of the funding plan, not a consequence of duration matching.
Worked Example: Funding a Later Payment by Reinvesting an Interim Receipt
A hypothetical liability portfolio requires $121 at the end of year 2 and $266.20 at the end of year 3. The annual effective valuation and reinvestment rate is 10%. Asset B costs $100 per unit and pays $121 at the end of year 2. Asset C costs $100 per unit and pays $133.10 at the end of year 3. A portfolio holds two units of B and one unit of C. Determine its base present value, compare its duration with the liability duration, and test whether it can make both payments if receipts can be reinvested at 10%.
Base present values. The liability present value is
Its Macaulay duration is
The asset portfolio costs \(2(100)+1(100)=300\). Its cash flows are $242 at year 2 and $133.10 at year 3, so its present-value components are \(242/1.21=200\) and \(133.10/1.331=100\). Its duration is
Thus the portfolio matches present value but does not match duration: its duration is \(1/3\) year shorter. It also does not have the same cash flows as the liabilities on each date.
Cash-flow test. At year 2, the assets pay $242 and the liability payment is $121. Pay the liability and reinvest the remaining \(242-121=121\) for one year. At 10%, that reinvestment grows to \(121(1.10)=133.10\) at year 3. Add the year-3 payment from asset C:
The year-3 liability can therefore be paid exactly under the stated reinvestment assumption. The plan has no year-2 shortfall, and the year-3 amount checks exactly. The common mistake is to treat the year-2 surplus as if it were already worth $121 at year 3; it grows to $133.10 only because the reinvestment rate and one-year period have been specified.
This example also shows why a successful cash-flow plan and a duration match are different tests. The reinvestment plan works under its stated assumptions, even though the portfolio’s base duration does not equal the liability duration. Conversely, equal present value and duration would not guarantee that a payment-date shortfall can be avoided.
Nonparallel Stress Tests With Holding Constraints
Duration is useful for comparing interest-rate sensitivity around a base valuation, but a nonparallel yield change can move different maturities by different amounts. For an integrated selection, use the actual stressed values supplied by each scenario, calculate the portfolio surplus in each one, and respect all holding limits. A scenario surplus is asset value minus liability value; a negative result is a shortfall under that scenario.
Worked Example: Choosing the Best Feasible Duration-Matched Portfolio
At a 5% annual effective base valuation yield, assets A, B, and C each have base present value $100 per unit and durations 1, 2, and 3 years, respectively. The liability present value is $300 and its duration is 2 years. Holding constraints are \(x_A\geq0.5\), \(x_C\leq0.75\), and \(x_B\leq2\), with all holdings nonnegative. The year-1 liquidity requirement is represented by the minimum holding of A. Three nonparallel stress scenarios give the following market values per unit and liability values; these stressed values are supplied inputs.
| Scenario | A value | B value | C value | Liability value |
|---|---|---|---|---|
| Short-end increase | $98 | $97 | $96 | $294 |
| Long-end increase | $99 | $97 | $94 | $291 |
| Twist | $96 | $98 | $99 | $297 |
Select feasible holdings that maximize the smallest surplus across the three scenarios, while matching base present value and duration.
Step 1: Express the base matching equations. Present-value matching requires
The liability dollar-duration amount is \(300(2)=600\) dollar-years, so duration matching requires
Subtracting the first equation from the second gives \(x_B+2x_C=3\). Combining this with the value equation gives \(x_A=x_C\). Set \(x_A=x_C=t\); then \(x_B=3-2t\).
Step 2: Apply the holding limits. The constraints imply \(t\geq0.5\), \(t\leq0.75\), and \(3-2t\leq2\), or \(t\geq0.5\). Thus the feasible range is \(0.5\leq t\leq0.75\), and \(x_B\) remains nonnegative throughout.
Step 3: Calculate each stressed surplus. For the short-end increase, asset value is
so surplus is \(291-294=-3\), independent of \(t\). For the long-end increase, asset value is \(99t+97(3-2t)+94t=291-t\), giving surplus \(-t\). For the twist, asset value is \(96t+98(3-2t)+99t=294-t\), giving surplus \(-3-t\).
Step 4: Optimize and check. Throughout \(0.5\leq t\leq0.75\), the twist surplus \(-3-t\) is smaller than both \(-3\) and \(-t\). Maximizing the smallest surplus therefore means choosing the smallest feasible \(t\), which is \(t=0.5\). The selected holdings are \(x_A=0.5\), \(x_B=2\), and \(x_C=0.5\).
At these holdings, stressed asset values are $291, $290.50, and $293.50. The corresponding surpluses are \(-$3, -$0.50, and \(-$3.50\); the worst is \(-$3.50\). The base checks are present value \(100(0.5+2+0.5)=$300\) and dollar duration \(100(0.5+4+1.5)=$600\). The holdings satisfy every stated limit. The common mistake is to select a portfolio from the base duration equations alone and assume that this also gives the best nonparallel stress result; the scenario calculations show which feasible choice actually improves the worst surplus.
A Reliable Integrated Workflow
The order of operations matters. A portfolio can meet its base matching targets but fail a holding limit, require unavailable cash before a liability date, or perform poorly under a specified stress. Use a separate calculation for each test rather than treating any one result as a substitute for the others.
Record the rate or valuation information, the present values and durations being used, and any assumptions about deterministic cash flows and reinvestment.
Match total present value and dollar duration. Use consistent units, such as dollars and dollar-years, throughout.
Apply nonnegativity, fixed holdings, upper bounds, liquidity needs, and other stated restrictions. An algebraic solution with a negative holding is not a feasible portfolio under a no-short-sale constraint.
For every liability date, compare the cash available then with the payment due. Identify any interim surplus, shortfall, or reinvestment assumption explicitly.
Use the scenario values for assets and liabilities, calculate surplus separately for each scenario, and optimize only over portfolios that remain feasible.
This integrated approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The central discipline is to keep distinct quantities distinct: base present value, dollar duration, cash available on a payment date, and stressed surplus answer different questions.
Check Your Understanding
For each problem, use the stated assumptions and check both the matching equations and any feasibility or cash-flow condition.
- Two assets each have present value $500 per unit and durations 1 and 3 years. A liability has present value $1,500 and duration 2 years. Find nonnegative holdings that match present value and duration. (Answer: 1.5 units of each asset.)
- A liability of $1,070 is due at the end of year 2. A portfolio receives $1,000 at the end of year 1, with no other cash before the liability date. If the annual effective reinvestment rate is 4%, how much cash is available at year 2, and what is the shortfall? (Answer: $1,040 available; $30 shortfall.)
- Asset A has present value $800 and duration 1.5 years per unit; asset B has present value $1,200 and duration 3 years per unit. A liability has present value $2,000 and duration 2.4 years. Find holdings that match both targets. (Answer: 1 unit of A and 1 unit of B.)
- A duration-matched portfolio has two units of asset A and one unit of asset B. Under a stated stress, their per-unit values are $96 and $103, respectively, while the liability value is $295. Calculate the stressed surplus. (Answer: $0.)
- For the scenario-optimization example in this tutorial, suppose an additional limit changes to \(x_C\leq0.6\), with all other data unchanged. What is the feasible interval for \(t\), and which value maximizes the worst scenario surplus? (Answer: \(0.5\leq t\leq0.6\); choose \(t=0.5\), giving holdings \(x_A=0.5,\ x_B=2,\ x_C=0.5\).)