What It Means to Match a Single Liability at a Horizon
Suppose an investor must make one known payment on a specified future date. The target horizon is the time from today to that payment. The funding question is not merely whether the assets have an appropriate value today: it is whether the assets will provide enough value at the horizon, under the way they are held and managed. Interim asset payments may need to be reinvested; an asset that has not matured may need to be sold.
A comparison at today’s valuation rate provides a useful starting point. It does not, by itself, guarantee that the liability will be funded if rates change or interim cash flows earn a different reinvestment rate. The horizon calculation makes those assumptions explicit.
If the asset is still outstanding at \(T\), its value at the horizon must also be included if the plan is to sell it or otherwise use its market value to pay the liability. That value depends on the market yield at \(T\) and the asset’s remaining cash flows. The formula for accumulated proceeds above covers only cash flows already received by the horizon.
Present Value and Horizon Value at the Same Rate
For a fixed asset cash-flow stream whose payments occur no later than \(T\), its present value at rate \(i\) is \(P=\sum_{t\leq T} C_t(1+i)^{-t}\). If those same cash flows are reinvested to time \(T\) at that same rate, their accumulated value equals \(P(1+i)^T\). The single liability’s accumulated value at the same rate is also its present value multiplied by \((1+i)^T\).
The equality follows by factoring \((1+i)^T\) out of every term: \((1+i)^T(1+i)^{-t}=(1+i)^{T-t}\). Therefore, if the asset present value equals the liability present value at rate \(i\), then accumulating both at \(i\) gives equal values at \(T\). This is an accounting identity under the stated rate assumption, not a promise that the realized proceeds will match if reinvestment rates differ.
Duration provides a separate description of cash-flow timing. For known positive asset cash flows, the Macaulay duration at rate \(i\) is the present-value-weighted average payment time:
A single liability payable at \(T\) has duration \(T\). An asset with earlier cash flows will generally have duration below its final maturity, because those earlier payments receive positive weight in the average. Duration is useful for comparing timing, but it does not specify the cash amount available at the horizon or remove the need to state reinvestment and sale assumptions.
Worked Horizon Comparisons
Worked Example: A Zero-Coupon Asset Maturing on the Liability Date
A fund owes $12,100 in two years. The effective annual valuation rate is 10%. It can buy a zero-coupon asset paying $12,100 at the end of year 2 for $10,000. Compare the present values, durations and proceeds at the liability date. Assume the asset is held to maturity.
Method and assumptions. Discount the liability and asset payment at 10%. There are no interim asset payments, so no reinvestment assumption is needed. The asset’s maturity payment is certain under the example’s assumptions.
Present values. The liability present value is
The asset costs $10,000 today, so its present value equals the liability present value. Both present values are calculated at the same effective annual rate.
Duration. The asset has one positive cash flow at time 2, so its present-value-weighted average payment time is \(D=2\) years. The liability also has a single payment at year 2 and duration \(T=2\) years.
Horizon proceeds and check. The asset pays $12,100 at year 2, exactly the liability amount. Accumulating the equal present values for two years at 10% also checks the result: \(10{,}000(1.10)^2=\$12{,}100\). The match here relies on holding the asset to maturity and receiving its promised payment; selling before maturity would introduce a market-price question.
Common mistake avoided. Do not discount the liability for two years but compare that present value directly with the year-2 payment. Compare present values with present values, or accumulated values with accumulated values.
Worked Example: A Coupon Requiring Reinvestment
A fund owes $11,025 in two years. Its valuation rate is 5%. It buys a two-year bond for $10,000 that pays $500 at the end of year 1 and $10,500 at the end of year 2. Calculate its duration and its year-2 proceeds if the first payment is reinvested at 3%. Compare the proceeds with the liability.
Method and assumptions. First value both sides at the 5% valuation rate. Then accumulate the year-1 coupon for one year at the stated 3% reinvestment rate. The final bond payment is received at the liability date. Ignore default, transaction costs and taxes.
Present-value check. The bond’s present value is
The liability’s present value is \(11{,}025/(1.05)^2=11{,}025/1.1025=\$10{,}000\). Thus the asset and liability present values match at 5%.
Duration. The present-value weights are \(476.190476\ldots/10{,}000=1/21\) for the year-1 payment and \(9{,}523.809524\ldots/10{,}000=20/21\) for the year-2 payment. Therefore,
The liability duration is 2 years. The bond duration is slightly shorter because part of its present value comes from the earlier coupon.
Horizon proceeds. The $500 coupon earns \(500(1.03)=\$515\) by year 2. Adding the bond’s year-2 payment gives total proceeds of \(515+10{,}500=\$11{,}015\). The funding difference, defined as proceeds minus liability, is \(11{,}015-11{,}025=-\$10\). The fund is short by $10.
Check and lesson. If the coupon instead earned 5%, proceeds would be \(500(1.05)+10{,}500=\$11{,}025\), exactly the liability. The present-value match therefore guarantees the horizon equality only under reinvestment at the valuation rate in this calculation. Duration describes timing; it does not replace the explicit reinvestment calculation.
Common mistake avoided. Do not count the $500 year-1 coupon as if it were already $500 at year 2, and do not reinvest it at the 5% valuation rate when the problem specifies 3%.
Worked Example: An Asset That Must Be Sold at the Horizon
A fund owes $12,100 in two years. The effective annual valuation rate is 10%. It considers a zero-coupon asset paying $13,310 at the end of year 3. If the asset is sold at the end of year 2, calculate its market value when the yield for its one remaining year is 8% and when it is 12%. Compare each value with the liability.
Method and assumptions. First compare today’s present values at 10%. At year 2, value the remaining year-3 payment using the yield then applicable for that one year. Assume the sale can be made at that market value, with no transaction costs or credit losses.
Present values and timing. At 10%, the liability present value is \(12{,}100/(1.10)^2=\$10{,}000\). The asset present value is \(13{,}310/(1.10)^3=13{,}310/1.331=\$10{,}000\). Their present values match, but the asset pays at year 3, one year after the liability date. Its duration is 3 years; the liability duration is 2 years.
Sale value at an 8% yield. At year 2, one year remains until the asset payment. Its market value is
The funding difference is \(12{,}324.07-12{,}100=\$224.07\), a surplus.
Sale value at a 12% yield. The market value becomes
The funding difference is \(11{,}883.93-12{,}100=-\$216.07\), a shortfall. Values are rounded to the nearest cent only at the final step.
Check and lesson. The same initial present-value match produces a surplus or shortfall depending on the yield at which the remaining payment is valued at the horizon. Because the asset has not matured, the fund must sell it or find another source of cash to pay the liability.
Common mistake avoided. Do not treat the $13,310 year-3 payment as cash available at year 2. The amount available from a sale at year 2 is its then-current market value, not its future face payment.
What the Comparison Can and Cannot Establish
The examples illustrate three distinct funding situations. An asset that pays exactly at the liability date can provide a direct match under the assumed payment conditions. An asset with an earlier coupon requires a reinvestment assumption for that coupon. An asset with a later maturity requires either a sale at the horizon or another source of funds, so its horizon value depends on the market yield then prevailing.
Present value, duration and accumulated proceeds answer different questions:
| Measure | Question answered | Important limitation |
|---|---|---|
| Present value | What is the value today at the stated valuation rate? | A match at one rate does not specify realized reinvestment rates or future sale prices. |
| Duration | What is the present-value-weighted timing of the cash flows? | It is a timing summary, not a cash amount available at the horizon. |
| Accumulated proceeds | What amount is available at the horizon under the stated reinvestment and sale assumptions? | The result depends on those assumptions and on which payments have actually been received. |
A practical horizon analysis should identify the liability date and amount, list the asset cash flows and their dates, and state what happens to each cash flow before the horizon. Receipts before the liability date need a reinvestment rule. Payments after that date are not horizon cash unless the asset is sold or otherwise valued at the horizon.
This framework is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It is especially useful for catching a common analytical shortcut: concluding that two positions are fully matched because their present values are equal. Equality is informative, but a funding conclusion also requires attention to cash-flow timing and the route by which assets become cash at the liability date.
Common Horizon-Matching Errors
- Mixing dates. A present value today cannot be compared directly with a liability amount due later. Bring both values to the same date using stated assumptions.
- Ignoring interim cash flows. A coupon received before the horizon does not automatically earn the valuation rate. Its accumulated value depends on its actual reinvestment rate.
- Counting a later payment too early. If an asset matures after the liability date, use its market value at the horizon if it must be sold; do not count its later face payment as current cash.
- Treating duration equality as exact funding. Duration summarizes weighted timing. It does not establish that proceeds will equal the liability under all rate paths.
- Leaving assumptions unstated. A numerical horizon value is meaningful only after specifying reinvestment rates, whether the asset is held or sold, and the yield used to value any remaining payments.
Check Your Understanding
For each problem, state the relevant rate or sale assumption and compare amounts at the same date.
- A liability of $8,000 is due in two years. At an effective annual rate of 4%, find its present value and verify the accumulated value at year 2. (Answer: Present value \(=\$7{,}396.45\); accumulated value \(=\$8{,}000.00\).)
- A two-year bond costs $5,000 at a 6% valuation rate and pays $300 at year 1 and $5,300 at year 2. A liability of $5,618 is due at year 2. If the year-1 payment is reinvested at 4%, calculate the funding difference at year 2. (Answer: Proceeds \(=\$5{,}612\); shortfall \(=\$6\).)
- A liability of $12,100 is due in two years. An asset pays $12,100 at year 3 and is valued at 10% today. What is its present value, and what is its sale value at year 2 if the remaining one-year yield is 8%? (Answer: Present value \(=\$9{,}090.91\); year-2 sale value \(=\$11{,}203.70\).)
- A bond pays $200 at year 1 and $5,200 at year 2. At a 4% effective annual rate, its price is $5,000. Calculate its Macaulay duration and compare it with a liability due at year 2. (Answer: Bond duration \(=1.96154\) years; liability duration \(=2\) years.)