Two Different Meanings of “Matched”
An asset portfolio can be said to match a liability portfolio in more than one sense. The strongest form is exact cash-flow matching: the asset payments equal the liability payments at every date. A less restrictive approach matches the portfolios’ present values and durations at a chosen interest rate. That second match is useful, but it does not make the future cash flows identical.
The distinction matters because a present-value calculation compresses a schedule of payments into a single number, and duration summarizes one aspect of that schedule’s sensitivity to interest rates. Neither summary, by itself, confirms that cash will be available on every liability payment date. This tutorial compares what each method achieves and the assumptions needed to interpret the comparison.
What Each Match Controls
Suppose all payments occur at times \(t=0,1,\ldots,n\), are fixed, and are discounted at a common effective annual rate \(i\). Let \(v=1/(1+i)\). The present value of a portfolio with cash flow \(C_t\) at time \(t\) is \(\sum_t C_t v^t\). For positive cash flows, its Macaulay duration is the present-value-weighted average payment time.
For liabilities, \(P_L\) and \(D_L\) denote present value and duration; for assets, use \(P_A\) and \(D_A\). Present-value-and-duration matching requires \(P_A=P_L\) and \(D_A=D_L\) at the chosen rate. The comparison assumes fixed cash flows and a common flat rate for discounting. If cash flows depend on interest rates, or asset and liability values use different rate assumptions, this simple comparison does not by itself establish a match.
Worked Example: Exact Matching at Every Payment Date
A hypothetical liability portfolio pays $110 at the end of year 1 and $121 at the end of year 2. A default-free asset portfolio pays the same amounts on the same dates. At an effective annual rate of 10%, verify the present-value match and explain what happens to the net cash flows if the discount factors change.
Method. Compare payments date by date first, then calculate present values. This avoids treating equality of present values as proof of equality of cash-flow schedules.
Cash-flow comparison. At year 1, assets pay $110 and liabilities require $110, so the net cash flow is \(110-110=0\). At year 2, the net cash flow is \(121-121=0\). There are no other payments in either schedule.
Present values at 10%. The year-1 discount factor is \(1/1.10\), and the year-2 discount factor is \(1/1.10^2\). Thus
The asset and liability durations are also equal: each has present-value contributions of $100 at times 1 and 2, so each duration is \((1(100)+2(100))/200=1.5\) years. But exact matching is stronger than this calculation. For example, if common discount factors at years 1 and 2 were instead 0.95 and 0.90, each portfolio would have present value \(110(0.95)+121(0.90)=104.50+108.90=\$213.40\). At every date, the asset payment still exactly covers the liability payment.
Check and mistake avoided. The two schedules match date by date, not merely in total. A present-value equality at one interest rate alone would not establish this result.
Why Duration Matching Controls First-Order Rate Sensitivity
Duration matching has a precise but narrower purpose. With fixed cash flows discounted at one effective annual rate, differentiating present value with respect to that rate gives
The first equality follows by differentiating each term \(C_t(1+i)^{-t}\): its derivative is \(-tC_t(1+i)^{-(t+1)}\). For the second equality, factor out \(1/(1+i)\); the remaining sum is the duration numerator \(D(i)P(i)\).
Let the surplus value be \(S(i)=P_A(i)-P_L(i)\), with positive values indicating an excess of asset value. If the portfolios have equal present values and equal durations at \(i_0\), then \(S(i_0)=0\), and
Thus a small change in the common rate produces no first-order change in the surplus at the matching rate. This is a local statement: it does not assert that the surplus remains zero after a large rate change, or that cash flows match on each date. It also concerns a parallel change to one common rate, not arbitrary changes to different parts of the yield curve.
Worked Example: Verifying a Present-Value-and-Duration Match
At an effective annual rate of 10%, a hypothetical liability pays $120 at the end of year 1 and $145.20 at the end of year 3. An asset portfolio pays \(120/1.10\) immediately and $159.72 at the end of year 4. Verify whether present values and durations match, and determine whether the cash flows match exactly.
Method. Discount every payment to time 0, then use present-value contributions to calculate duration. This makes the timing of the immediate asset payment explicit.
Liability present value. The discount factors are \(1/1.10\) at year 1 and \(1/1.10^3=1/1.331\) at year 3. Therefore
The liability duration numerator is \(1(120/1.10)+3(145.20/1.331)=4(120/1.10)\). Consequently, \(D_L=4/2=2\) years.
Asset present value. The immediate payment is already at time 0, so its present value is \(120/1.10=\$109.0909\ldots\). Also, \(1.10^4=1.4641\), and \(159.72=120(1.10)^3\), so the year-4 payment has present value
It follows that \(P_A=2400/11=\$218.1818\ldots=P_L\). The asset duration numerator is \(0(120/1.10)+4(159.72/1.4641)=4(120/1.10)\), giving \(D_A=2\) years. The portfolios therefore match in present value and duration at 10%.
Are they exact cash-flow matches? No. The liability payments occur at years 1 and 3; the asset payments occur at time 0 and year 4. The matching summary measures do not align the dates or amounts of individual payments.
Check and mistake avoided. The year-4 asset payment is discounted by \(1.10^4\), not by \(1.10^3\). Using \(120(1.10)^4=\$175.692\) instead would give a year-4 present value of $120 and asset present value \(120/1.10+120=\$229.0909\ldots\), not $218.1818. Its duration would be \(4(120)/(229.0909\ldots)=44/21\approx2.0952\) years, so it would not match.
Risks Eliminated and Risks Left Exposed
The following comparison assumes fixed, certain asset and liability payments and, for duration matching, a common flat discount rate. Actual portfolios can introduce additional risks, such as default, transaction costs, or embedded options; neither matching calculation removes those risks automatically.
| Feature | Exact cash-flow matching | Present-value-and-duration matching |
|---|---|---|
| Payment dates and amounts | Assets cover liabilities at each matched date. | Schedules can differ even when present value and duration agree. |
| Interest-rate changes | Identical dated cash flows have equal values under any common discount factors. | Surplus has zero first-order sensitivity to a small parallel rate change at the matching rate. |
| Funding timing | Matched asset payments are available when the corresponding liabilities are due. | Aggregate duration does not guarantee cash is available on every liability date. |
| Reinvestment and cash balances | No reinvestment of matched payments is needed when each payment is used at the same date. | Earlier or excess asset payments may need to be held or reinvested to meet later liabilities. |
| Other changes | Does not remove default, transaction-cost, or cash-flow uncertainty risk. | Also leaves exposure to larger rate changes and changes that are not parallel. |
Exact matching is a date-by-date funding method, not just a valuation test. It can eliminate the rate sensitivity of the net scheduled cash flows when the asset and liability payments truly coincide and are equally certain. Duration matching is a sensitivity-control method: it can be less demanding in its cash-flow requirements, but its protection is local and depends on the rate-change assumptions.
Worked Example: A Duration Match That Is Not Exact
At 10%, liabilities pay $110 at the end of year 1 and $121 at the end of year 2. Assets pay $100 immediately and $133.10 at the end of year 3. Verify the present-value-and-duration match. Then recalculate the surplus value at a 0% effective annual rate and decide whether the cash flows are exactly matched.
Method. First compare present-value contributions at 10%; then revalue both fixed schedules at 0%. This distinguishes a local match at one rate from equality of cash flows.
Liabilities at 10%. The present values are \(110/1.10=\$100\) and \(121/1.10^2=\$100\). Thus \(P_L=\$200\), and \(D_L=(1(100)+2(100))/200=1.5\) years.
Assets at 10%. The immediate $100 has present value $100. Since \(1.10^3=1.331\), the year-3 payment has present value \(133.10/1.331=\$100\). Therefore \(P_A=\$200\) and \(D_A=(0(100)+3(100))/200=1.5\) years. The portfolios match in present value and duration at 10%.
Revaluation at 0%. At zero interest, every discount factor is 1. The liability present value is \(110+121=\$231\). The asset present value is \(100+133.10=\$233.10\). The surplus is \(P_A-P_L=233.10-231=\$2.10\).
The nonzero surplus after this rate change confirms that duration matching is not an exact match. In contrast, if the assets paid $110 at year 1 and $121 at year 2, the surplus would be zero at 0% and at any other common discount rate.
Check and mistake avoided. Matching both summary measures at 10% does not justify carrying the equality of present values over to a different rate. The two schedules must be revalued using the new rate.
Common Exam Traps
- Calling a duration match an exact match. Check the cash flows at each date; equal present values and durations do not establish equal payment schedules.
- Omitting the payment time from the duration numerator. A payment at time 0 contributes zero to the numerator, even though it contributes its full amount to present value.
- Using different discount conventions for the two portfolios. A comparison requires consistent rates and timing assumptions.
- Overstating the interest-rate conclusion. Equal present values and durations remove first-order sensitivity to a small parallel change under the fixed-cash-flow model. They do not guarantee equal values after arbitrary rate changes.
- Ignoring funding dates. An aggregate present-value match cannot by itself show that cash is available on each liability due date.
Check Your Understanding
For each problem, identify whether the stated information establishes exact cash-flow matching, present-value-and-duration matching, both, or neither.
- Assets and liabilities each pay $260 at the end of year 2 and have no other payments. At an effective annual rate of 7%, state whether the cash flows match exactly and whether their present values and durations match. (Answer: Both; each present value is \(260/1.07^2\approx\$227.04\), and each duration is 2 years.)
- At 5%, liabilities pay $105 at year 1 and $121 at year 2. Calculate their present value and Macaulay duration. (Answer: \(P_L=\$209.75\), \(D_L=319.5/209.75\approx1.5232\) years.)
- At 10%, liabilities pay $110 at year 1 and $121 at year 2. Assets pay $100 immediately and $133.10 at year 3. Are the portfolios exactly matched, or do they match in present value and duration? (Answer: They are not exactly matched; both have present value $200 and duration 1.5 years at 10%.)
- At 8%, assets and liabilities each have present value $500. Their durations are 3.8 years and 4.0 years, respectively. For surplus \(S=P_A-P_L\), calculate \(S'\) at the matching rate and estimate the surplus change for a small parallel rate increase of 1 percentage point. (Answer: \(S'=-(500(3.8)-500(4.0))/1.08\approx+\$92.5926\) per unit rate; the first-order estimated change is about \(+\$0.93\).)
- Two portfolios have identical cash flows at every date, but the discount factors change differently by date. Does their equality of present value persist? (Answer: Yes, provided both portfolios are valued with the same discount factor at each date; equal dated cash flows produce equal present values under any common set of discount factors.)