Why Matching Errors Matter
An asset-liability match is only meaningful when the two sides are measured on compatible terms. A reported surplus can be misleading if asset and liability values refer to different dates, use inconsistent interest assumptions, or ignore when cash is actually available. Even a carefully constructed duration match has limits: it describes a local response to a specified type of interest-rate movement, not equality of cash flows or protection against every change in rates.
The focus here is on diagnosing those limitations. The calculations use annual cash flows and, unless otherwise stated, a flat annual effective interest rate. A flat rate is an assumption for the example, not a claim that market rates are always flat. When a stress scenario supplies discount factors directly, use those stated factors for that scenario rather than assuming the original rate still applies.
Check the Valuation Date First
A value at one date cannot be subtracted from a value at another date to obtain an economically meaningful surplus. Under a constant annual effective rate \(i\), an amount \(C\) available in one year has value \(C/(1+i)\) today, and an amount \(V\) today accumulates to \(V(1+i)\) in one year. These transformations let us restate both sides at one common date.
Worked Example: Comparing Values at a Common Date
A fund has assets worth $9,500 at time 0. A liability of $10,000 is due at time 2. Assume a constant annual effective rate of 5%. Find the surplus at time 0 and at time 1, and explain why subtracting the time-1 liability value from the time-0 asset value is invalid.
Method. First put both values at time 0. Then accumulate both values to time 1 to verify that the surplus is being carried consistently.
At time 0, the liability value is \(10{,}000/(1.05)^2=10{,}000/1.1025=9{,}070.2947846\). The asset value is already stated at time 0, so \(9{,}500-9{,}070.2947846=429.7052154\). The time-0 surplus is $429.71 to the nearest cent.
At time 1, the assets accumulate to \(9{,}500(1.05)=9{,}975\). The liability is one year away, so its time-1 value is \(10{,}000/1.05=9{,}523.8095238\). The time-1 surplus is \(9{,}975-9{,}523.8095238=451.1904762\), or $451.19 to the nearest cent.
Check. The time-0 surplus accumulates one year at 5%: \(429.7052154(1.05)=451.1904762\), which agrees with the time-1 calculation. The invalid subtraction would be \(9{,}500-9{,}523.8095238\): it mixes a time-0 asset value with a time-1 liability value. The error is not repaired by labeling the resulting difference a surplus.
In more involved settings, the same audit applies even if amounts are described as “current,” “projected,” or “at settlement.” Establish the valuation date of every number before comparing or adding it. If assumptions for accumulation differ across periods, apply the assumptions appropriate to each interval rather than multiplying by a rate for the wrong period.
Use Consistent Discount Assumptions
Discounting assets at one rate and liabilities at another can manufacture an apparent surplus or deficit. There may be a justified reason to value assets and liabilities under different prescribed assumptions, but the difference must be intentional and reported as such. It is not a valid like-for-like comparison merely because both calculations are present values.
Worked Example: An Apparent Surplus From Mixed Rates
A portfolio of assets pays $100 at time 1 and $110 at time 2. The liabilities have exactly the same payments on the same dates. An analyst discounts the assets at 4% and the liabilities at 5%. Calculate the apparent surplus, then compare the values using one common rate.
Method. Value each cash-flow stream on the same date, keeping full precision in intermediate calculations. Then identify whether any difference remains when both streams use the same rate.
Using 4% for the assets gives \(100/1.04=96.1538461538\) and \(110/(1.04)^2=110/1.0816=101.7011834319\). Thus the asset value is \(96.1538461538+101.7011834319=197.8550295857\).
Using 5% for the liabilities gives \(100/1.05=95.2380952381\) and \(110/(1.05)^2=110/1.1025=99.7732426304\). Thus the liability value is \(95.2380952381+99.7732426304=195.0113378685\). The reported difference is \(197.8550295857-195.0113378685=2.8436917172\), an apparent surplus of $2.84 to the nearest cent.
But the asset and liability cash flows are identical. At a common 4% rate, both values are $197.8550295857; at a common 5% rate, both are $195.0113378685. The surplus is zero on either common basis. The $2.84 difference is entirely due to applying different discount rates to the two sides.
Check and trap avoided. The result follows from identical payments receiving identical values when valued on the same basis. Rounding each payment’s present value before summing can also create a small arithmetic discrepancy. Keep unrounded values through the sum and round only the final reported amount.
Duration Is Not Cash-Flow Equality
For cash flows valued at a single annual effective rate \(i\), the present value is the sum of each payment discounted to the valuation date. Macaulay duration is the present-value-weighted average payment time. Equal present values and equal durations imply a first-order match for a small parallel change in the rate under this model. They do not imply that payments occur on the same dates or in the same amounts.
To see why the derivative result holds, write the present value as a function of the rate \(y\): \(V(y)=\sum_t C_t(1+y)^{-t}\). Differentiating each term gives \(V'(y)=-\sum_t tC_t(1+y)^{-t-1}\). At \(y=i\), this is \(-VD/(1+i)\). Therefore two portfolios with the same \(V\) and \(D\) have the same first derivative at \(i\). The conclusion concerns the local slope only; later rate movements, cash-flow timing, and curvature can still produce different outcomes.
Worked Example: Equal Duration but Different Cash Flows
A liability pays $100 at time 2. Assets pay \(1000/21\) dollars at time 1 and $52.50 at time 3. Assume a flat annual effective rate of 5%. Verify that present values and durations match. Then apply a nonparallel stress whose supplied discount factors at times 1, 2, and 3 are 0.95, 0.90, and 0.85, respectively.
Method. First calculate present values and duration using the original flat 5% basis. For the stress, use the specified factors at the respective dates; do not infer that the original flat-rate assumption still describes the stressed values.
The asset payment at time 1 is \(1000/21=47.6190476190\). Its present value is \((1000/21)/1.05=45.3514739229\). The time-3 asset payment has present value \(52.50/(1.05)^3=52.50/1.157625=45.3514739229\). The asset present value is therefore \(45.3514739229+45.3514739229=90.7029478458\).
The liability present value is \(100/(1.05)^2=100/1.1025=90.7029478458\), so the values match. The asset duration is the weighted average of times 1 and 3 with equal present-value weights: \((1(45.3514739229)+3(45.3514739229))/90.7029478458=2\). The liability is paid entirely at time 2, so its duration is also 2.
Under the stressed discount factors, the asset value is \((1000/21)(0.95)+52.50(0.85)=45.2380952381+44.625=89.8630952381\). The liability value is \(100(0.90)=90\). The stressed surplus is \(89.8630952381-90=-0.1369047619\), a deficit of $0.14 to the nearest cent.
Check and trap avoided. The original values and durations match exactly, but the cash flows do not: the assets pay at times 1 and 3, while the liability is due at time 2. The stress changes discount factors differently across dates, so the parallel-change duration result does not ensure a match. Treating equal duration as proof of cash-flow equality—or as protection from a curve twist—is the mistake.
Present-Value Surplus Does Not Guarantee Interim Liquidity
A portfolio can have assets whose total present value exceeds the liabilities while still lacking cash when a payment is due. A later asset receipt may be available to meet an earlier obligation only if the fund can borrow, sell assets, or draw on another liquid resource. Those actions can have costs or constraints. Present value summarizes values at a date; it does not by itself establish that cash is available on every intervening date.
Worked Example: A Positive Surplus With an Earlier Cash Shortfall
Assets pay $60 at time 1 and $50 at time 2. A liability of $100 is due at time 1. Assume a constant annual effective rate of 5%. Find the time-0 present-value surplus and the time-1 funding gap. If the fund borrows the gap at 5% until time 2, determine the amount due then and the cash left after the time-2 asset payment.
Method. Value both streams at time 0 for the surplus, then separately compare cash receipts and payments at each due date.
The asset present value is \(60/1.05+50/(1.05)^2=57.1428571429+45.3514739229=102.4943310658\). The liability present value is \(100/1.05=95.2380952381\). The time-0 surplus is \(102.4943310658-95.2380952381=7.2562358277\), or $7.26 to the nearest cent.
At time 1, the fund receives $60 but owes $100, so its immediate funding gap is \(100-60=40\). If it borrows $40 for one year at 5%, it owes \(40(1.05)=42\) at time 2. The $50 asset payment at time 2 repays that amount and leaves \(50-42=8\), before any other costs or payments.
Check and trap avoided. The calculations show both facts at once: the present-value surplus is positive, and the fund still needs $40 of cash at time 1. Assuming that a positive surplus automatically means every liability can be paid when due ignores timing and liquidity.
A Practical Audit for Matching Analyses
Before relying on a reported match, check what the analysis actually establishes. A match may target present value, duration, convexity, cash-flow timing, selected key-rate exposures, or some combination. These are different objectives. The fact that one objective is met does not establish the others.
Restate every asset and liability value at the same date, using the appropriate accumulation or discounting for the elapsed time.
Record the rate or other valuation basis used for each side. If they differ, determine whether that difference is required and describe the resulting comparison accurately.
List receipts and payments by date. Ask whether assets pay in time to fund each liability without relying on an unstated borrowing or sale.
Equal duration supports a first-order comparison for small parallel rate changes under the stated model. It does not establish cash-flow equality or protection against nonparallel movements.
Apply the scenario’s assumptions at each relevant date and calculate the resulting surplus. Do not extend a local sensitivity conclusion beyond the scenario it describes.
These checks matter in practice because a polished summary number can hide a mismatch in timing or assumptions. A report should state the valuation date, the rate basis, the matching objective, and any liquidity or scenario limitations. In particular, a duration match should be described as a sensitivity match under specified conditions, not as a guarantee that asset cash flows will meet liability cash flows.
- Date mismatch: a present value at one date is compared with an amount at another date.
- Basis mismatch: asset and liability values use different rates without explaining why.
- Duration overclaim: equal duration is treated as equal payments or as a guarantee under large rate moves.
- Interim cash omission: future receipts are assumed available for earlier payments without identifying a funding source.
- Curve-twist overclaim: a match for parallel shifts is treated as protection against changes that differ by maturity.
Check Your Understanding
For each problem, identify the relevant matching issue and show enough calculation to verify the result.
- An asset is worth $8,000 at time 0. A liability of $9,261 is due at time 2. At a constant annual effective rate of 5%, find the time-0 surplus and the time-1 surplus. (Answer: Time-0 surplus = \(-\$400.00\); time-1 surplus = \(-\$420.00\).)
- Assets and liabilities each pay $80 at time 1 and $120 at time 2. An analyst discounts assets at 3% and liabilities at 4%. Is the resulting difference evidence of an economic cash-flow surplus? State the corrected comparison if both are valued at 3%. (Answer: No; it is caused by inconsistent rates. At a common 3% rate, the identical cash flows have equal values and surplus is $0.)
- A liability pays $200 at time 2. Assets pay \(2000/21\) at time 1 and $105 at time 3. At a flat annual effective rate of 5%, verify whether their present values and Macaulay durations match, and state whether their cash flows match. (Answer: Both present values are \(181.4058957\), both durations are 2 years, but the dated cash flows do not match.)
- Assets pay $40 at time 1 and $70 at time 2; a liability of $100 is due at time 1. At a constant annual effective rate of 5%, find the time-0 surplus, the time-1 cash gap, and the time-2 remainder if the gap is borrowed at 5% for one year. (Answer: Surplus = $6.35; time-1 gap = $60; time-2 borrowing repayment = $63 and remainder = $7.)
- An asset portfolio has cash flows of \(1000/21\) at time 1 and $52.50 at time 3; the liability is $100 at time 2. Under stress, discount factors are 0.95, 0.90, and 0.85 at times 1, 2, and 3. Find the stressed surplus. Does the original duration match prevent this result? (Answer: Stressed surplus = \(-\$0.14\) to the nearest cent; no, duration matching under a parallel-rate model does not prevent a surplus change under a nonparallel stress.)