Partial Matching and Residual Cash Flows

Why Measure Residual Cash Flows?

A portfolio may cover some liability payments exactly without replicating every liability cash flow. At other dates, asset payments may exceed liabilities, while at still others they may fall short. Comparing only total present values can conceal this dated pattern. To assess what remains unmatched, calculate the residual at each payment date and then value those residuals using the stated discount curve.

This analysis is useful when exact cash-flow matching is infeasible or when a portfolio is deliberately only partially matched. It identifies both when cash flows are mismatched and the net present value of the mismatches as of the valuation date. A net surplus at the valuation date does not, by itself, mean that every liability can be paid from cash flows available on its due date.

Definition: At date $t_j$, let $A_j$ be the amount of asset cash flow and $L_j$ the amount of liability cash flow, with both amounts recorded as nonnegative magnitudes. Define the dated residual cash flow by $R_j=A_j-L_j$. A positive residual is an excess of asset cash flow over liability cash flow; a negative residual is a shortfall at that date.

Use the same valuation date, currency, and date units for all amounts. If assets and liabilities have cash flows on different dates, list the union of all those dates and enter zero for a side with no cash flow on a particular date. This preserves the timing information instead of forcing cash flows from different dates into one total.

Discounting the Residuals

Let $D(0,t_j)$ denote the discount factor, as of time 0, for a payment at time $t_j$. It must come from the stated valuation curve and be expressed using the same date convention as the cash flows. When the curve is specified by effective annual spot rates $i_{t_j}$, with annual compounding and $t_j$ measured in years, $D(0,t_j)=(1+i_{t_j})^{-t_j}$. If discount factors are supplied directly, use those factors rather than introducing a different rate assumption.

Valuation result: The net present value of the residual cash flows is $$ \text{PV}(R)=\sum_{j=1}^{n}D(0,t_j)R_j. $$ A positive result is a net present-value surplus; a negative result is a net present-value shortfall. This is a valuation of the signed residuals, not a statement that cash is available at every payment date.

The distinction between signed and absolute residuals matters. Positive and negative residuals can offset in the net present value. If it is useful to report the size of the dated mismatches without allowing opposite signs to cancel, a separate diagnostic is the discounted absolute residual sum, $\sum_j D(0,t_j)|R_j|$. This is not the net surplus or shortfall: it deliberately removes the sign before summing.

Why the Residual Valuation Equals the Difference in Values

The valuation identity follows from linearity. Assume the asset and liability cash flows are deterministic amounts at the listed dates, and that both are valued using the same discount factors. Their present values are the sums of their dated amounts multiplied by those factors. Subtracting the liability value from the asset value gives:

\begin{aligned} \text{PV}(A)-\text{PV}(L) &=\sum_{j=1}^{n}D(0,t_j)A_j-\sum_{j=1}^{n}D(0,t_j)L_j\\ &=\sum_{j=1}^{n}D(0,t_j)(A_j-L_j)\\ &=\sum_{j=1}^{n}D(0,t_j)R_j\\ &=\text{PV}(R). \end{aligned}

The identity requires consistent dates and discount factors. It does not require the asset and liability cash flows to match at each date. It does require the amounts being compared to be measured on a consistent basis; for example, one side must not be discounted with annual compounding while the other uses a different curve or valuation date.

List the dates.
Include every date on which either an asset or a liability cash flow occurs. Keep the original timing of each payment.

Find the dated residuals.
At each date, subtract the liability amount from the asset amount: $R_j=A_j-L_j$.

Apply the curve.
Multiply each residual by the discount factor for its own date, not by a factor for another date or by a single average factor.

Interpret the result.
Sum the signed discounted residuals for the net present-value surplus or shortfall. Inspect the dated residuals separately to see where cash flows are under- or over-covered.

Worked Applications

Worked Example: Residuals Across Three Payment Dates

A hypothetical fund has asset cash flows of $90, $75, and $60 at the ends of years 1, 2, and 3. Its liability cash flows on those dates are $100, $100, and $45. The valuation curve supplies discount factors $D(0,1)=0.97$, $D(0,2)=0.92$, and $D(0,3)=0.86$. Calculate each residual and the net present-value surplus or shortfall. All amounts are in dollars, with no mortality or other contingent cash flows.

Method. Calculate asset minus liability at each date, then discount each signed residual using that date’s factor. The sign convention avoids mistaking an excess asset payment for a shortfall.

At year 1, $R_1=90-100=-10$. At year 2, $R_2=75-100=-25$. At year 3, $R_3=60-45=15$. Thus the dated residuals are a $10 shortfall, a $25 shortfall, and a $15 excess, respectively.

The discounted residuals are $(-10)(0.97)=-9.70$, $(-25)(0.92)=-23.00$, and $(15)(0.86)=12.90$. Their sum is $-9.70-23.00+12.90=-19.80$, so the net present-value result is a $19.80 shortfall.

Check using separate present values. The asset value is $90(0.97)+75(0.92)+60(0.86)=87.30+69.00+51.60=\$207.90$. The liability value is $100(0.97)+100(0.92)+45(0.86)=97.00+92.00+38.70=\$227.70$. Their difference is $207.90-227.70=-\$19.80$, agreeing with the residual calculation.

Common mistake avoided: Adding the three residual amounts without discounting would give $-10-25+15=-\$20$, which is not their present value. Each dated amount has its own factor.

Worked Example: Comparing Asset and Liability Present Values

A hypothetical institution has asset cash flows of $120, $45, and $80 at the ends of years 1, 2, and 3. Its liability cash flows are $100, $70, and $90. The corresponding discount factors are 0.96, 0.90, and 0.83. Calculate the residuals, both present values, and the net shortfall. Cash flows are certain dollar amounts under the stated valuation curve.

Method. First compare the two cash flows date by date. Then value the residuals and independently calculate the asset and liability values as a check.

The residuals are $R_1=120-100=20$, $R_2=45-70=-25$, and $R_3=80-90=-10$. There is a $20 excess at year 1, followed by shortfalls of $25 and $10 at years 2 and 3.

The asset present value is $120(0.96)+45(0.90)+80(0.83)=115.20+40.50+66.40=\$222.10$. The liability present value is $100(0.96)+70(0.90)+90(0.83)=96.00+63.00+74.70=\$233.70$.

The residual present value is $20(0.96)+(-25)(0.90)+(-10)(0.83)=19.20-22.50-8.30=-\$11.60$. Therefore there is a $11.60 net present-value shortfall. The independent check is $222.10-233.70=-\$11.60$, as required.

Common mistake avoided: The year-1 surplus does not make the other dated residuals disappear. The net present value combines their values, but the dated list still shows shortfalls at years 2 and 3.

Worked Example: Net Shortfall Despite an Intermediate Surplus

A hypothetical portfolio produces asset cash flows of $85, $100, and $60 at the ends of years 1, 2, and 3, against liability cash flows of $100, $60, and $100. The valuation curve gives discount factors 0.95, 0.89, and 0.82 at those dates. Find the dated residuals and net present-value result. Treat all cash flows as certain and in dollars.

Method. Preserve the sign at each date. In particular, the large intermediate asset excess is discounted at the year-2 factor, not at a factor for year 1 or year 3.

At year 1, $R_1=85-100=-15$. At year 2, $R_2=100-60=40$. At year 3, $R_3=60-100=-40$. The portfolio has shortfalls of $15 and $40 at years 1 and 3, and an excess of $40 at year 2.

Discounting gives $(-15)(0.95)=-14.25$, $40(0.89)=35.60$, and $(-40)(0.82)=-32.80$. The total is $-14.25+35.60-32.80=-\$11.45$, a $11.45 net present-value shortfall.

Check from the two sides. Asset present value is $85(0.95)+100(0.89)+60(0.82)=80.75+89.00+49.20=\$218.95$. Liability present value is $100(0.95)+60(0.89)+100(0.82)=95.00+53.40+82.00=\$230.40$. The difference is $218.95-230.40=-\$11.45$, matching the residual valuation.

Common mistake avoided: The intermediate surplus is not the answer by itself. It offsets only the discounted shortfalls when calculating net present value; it does not turn the year-1 or year-3 dated shortfall into a surplus.

Interpreting the Result and Avoiding Traps

A signed residual present value answers a valuation question: how much greater is the present value of the asset cash flows than that of the liability cash flows under the stated curve? It does not answer whether assets deliver enough cash on every liability date. For that, inspect $R_j$ date by date. A negative $R_j$ identifies a dated cash-flow shortfall, even if the total residual present value is positive.

The discounted absolute residual sum can help describe the scale of dated mismatches without cancellation. For example, residuals of $+\$15,-\$25,+\$20$, discounted by 0.98, 0.93, and 0.87, have signed present value $15(0.98)-25(0.93)+20(0.87)=14.70-23.25+17.40=\$8.85$. Their discounted absolute residual sum is $15(0.98)+25(0.93)+20(0.87)=14.70+23.25+17.40=\$55.35$. The first is a net surplus; the second is a nonnegative measure of gross discounted mismatch. They answer different questions.

Common errors include reversing the residual sign, pairing a cash flow with the wrong discount factor, and interpreting a positive net value as proof of payment-date coverage. Another trap is confusing a discount factor with a discount rate: the factor multiplies the dated cash flow directly. If the problem supplies the curve’s factors, there is no need to reconstruct rates or assume a flat curve.

Key takeaway: Calculate $R_j=A_j-L_j$ separately at every date, discount each residual with its own stated factor, and sum the signed values for the net present-value surplus or shortfall. Keep the dated residuals visible: a net value can conceal individual shortfalls.

Check Your Understanding

For each problem, use the stated discount factors and the convention $R_j=A_j-L_j$. Amounts are certain dollar cash flows; no mortality or other contingencies are assumed.

At years 1, 2, and 3, assets pay $50, $35, and $40; liabilities pay $45, $35, and $50. The discount factors are 0.98, 0.94, and 0.90. Give the dated residuals and the net present-value result. (Answer: residuals $+\$5,\$0,-\$10$; net present-value shortfall of $4.10.)
Residuals at years 1, 2, and 3 are $+\$20,-\$25,-\$10$. The discount factors are 0.96, 0.90, and 0.83. Find the net present-value result. (Answer: net present-value shortfall of $11.60.)
Assets pay $85, $100, and $60 at years 1, 2, and 3; liabilities pay $100, $60, and $100. The discount factors are 0.95, 0.89, and 0.82. State the residuals and net present-value result. (Answer: residuals $-\$15,+\$40,-\$40$; net present-value shortfall of $11.45.)
Residuals at years 1, 2, and 3 are $+\$15,-\$25,+\$20$, with discount factors 0.98, 0.93, and 0.87. Find both the signed residual present value and the discounted absolute residual sum. (Answer: signed residual present value of $8.85 surplus; discounted absolute residual sum of $55.35.)
At years 1, 2, and 3, assets pay $70, $40, and $90, while liabilities pay $60, $65, and $70. The discount factors are 0.97, 0.91, and 0.84. Find the dated residuals and net present-value result. (Answer: residuals $+\$10,-\$25,+\$20$; net present-value surplus of $3.75.)

What You'll Learn