Why Asset-Liability Management Uses More Than One Measure
An insurer’s assets and liabilities create two related but distinct obligations: assets must provide cash when claims and benefits are due, and their market values must respond appropriately to changes in financial markets. A portfolio can have a positive market-value surplus yet lack cash at a required payment date. It can also have asset and liability values that are initially equal but move differently when rates change.
Three approaches address these concerns from different angles. Cash flow matching aligns asset receipts with liability payments by date. Duration matching seeks to reduce the first-order effect of a small parallel interest-rate change on market-value surplus. Surplus management monitors the difference between asset and liability values and assesses how that difference behaves under relevant scenarios. These approaches are complementary, not interchangeable.
Cash Flow Matching and Present Values
Suppose an insurer has deterministic asset cash flows \(C_t^A\) and liability cash flows \(C_t^L\) at dates \(t\). Cash flow matching seeks asset receipts at each relevant date that are at least as large as the liability payments due then. Exact matching means \(C_t^A=C_t^L\) at every payment date. In practice, cash flows may not match exactly, and an insurer may use liquid assets or other funding sources to cover gaps.
For a simplified valuation with annual compounding and a spot rate \(s_t\) applying to maturity \(t\), the present value of a cash flow at that date is its amount divided by \((1+s_t)^t\). The examples below assume fixed, certain cash flows and use the stated spot rates to discount both assets and liabilities. They do not include default, transaction costs, taxes, or uncertain claim amounts.
When \(C_t^A=C_t^L\) for each \(t\), every term in the sum is zero. This result depends on matching the amount, date, and relevant valuation basis. It does not guarantee a match if liability cash flows change, an asset payment is delayed or defaults, or the asset and liability cash flows are exposed to different currencies or discounting assumptions.
Worked Example: Repricing a Timing Mismatch After a Rate Shock
A hypothetical insurer holds assets paying $52 million at year 1 and $58.8 million at year 3. It owes $105.84 million at year 2. Initially, the annual spot rate is 5% at all three maturities. In a stress, only the 1-year spot rate rises to 6%; the 2-year and 3-year rates remain at 5%. Calculate initial and stressed market values and surplus. Report money in millions to three decimal places.
Method. Discount each cash flow using the spot rate for its own maturity. A common error is to apply the changed 1-year rate to every asset payment, even though the year-3 rate remains 5%.
Initial asset value. The year-1 receipt has present value \(52/1.05=49.523810\). The year-3 receipt has present value \(58.8/(1.05)^3=58.8/1.157625=50.793651\). Thus \(A=49.523810+50.793651=100.317461\) million.
Initial liability value. \(L=105.84/(1.05)^2=105.84/1.1025=96.000000\) million. Initial surplus is \(100.317461-96.000000=4.317461\) million.
Stressed asset value. The year-1 receipt is now worth \(52/1.06=49.056604\) million. The year-3 receipt remains worth \(58.8/(1.05)^3=50.793651\) million. Therefore, stressed assets are worth \(49.056604+50.793651=99.850255\) million.
Stressed liability value and surplus. The 2-year spot rate remains 5%, so liabilities are still worth \(105.84/(1.05)^2=96.000000\) million. Stressed surplus is \(99.850255-96.000000=3.850255\) million. Rounded to three decimal places, assets are $99.850 million, liabilities are $96.000 million, and surplus is $3.850 million. The stressed values are consistent with repricing only the cash flow whose maturity rate changed.
Cash Timing Is Not the Same as Market Value
Present values summarize the market value of future payments, but they do not move cash from one date to another. A receipt arriving after a claim payment cannot pay that claim on time unless the insurer has other funds or can borrow. Reinvestment can help with an earlier receipt, but the amount available later depends on the reinvestment rate and timing.
Worked Example: A Positive Surplus With a Payment-Date Shortfall
A hypothetical insurer owes $80 million at year 2 and holds assets paying $42 million at year 1 and $42 million at year 3. The annual spot rate is 5% at all maturities, with annual compounding. Assume the year-1 receipt can be reinvested at 5% to year 2. Find initial asset value, liability value, surplus, and the year-2 cash shortfall if no other funds are available. Report money in millions to three decimal places.
Method. First value each cash flow at the valuation date. Then separately identify cash actually available at year 2. The year-3 receipt contributes to market value but is not available at year 2.
Initial asset value. The year-1 asset is worth \(42/1.05=40.000000\) million. The year-3 asset is worth \(42/(1.05)^3=42/1.157625=36.281399\) million. Thus \(A=40.000000+36.281399=76.281399\) million.
Initial liability value and surplus. \(L=80/(1.05)^2=80/1.1025=72.562358\) million. Therefore, \(S=76.281399-72.562358=3.719041\) million.
Cash available at year 2. The year-1 receipt grows to \(42(1.05)=44.10\) million. The year-3 receipt has not yet arrived. The year-2 cash shortfall is \(80-44.10=35.90\) million. Rounded to three decimals, initial assets are $76.281 million, liabilities are $72.562 million, surplus is $3.719 million, and the year-2 shortfall is $35.900 million. A positive initial surplus does not remove the need to address the cash gap.
Duration Matching and Its Limits
Duration summarizes the timing of fixed cash flows in present-value terms. For a fixed-cash-flow portfolio valued at a single annual yield \(y\), its Macaulay duration is the present-value-weighted average payment time. Modified duration is the corresponding first-order measure of percentage price sensitivity to a small change in that yield. In a simple annual-compounding setting, \(D_{\text{mod}}=D_{\text{Mac}}/(1+y)\).
For a small parallel yield change \(\Delta y\), the first-order approximation is \(\Delta V\approx-D_{\text{mod}}V\Delta y\). Thus the approximate change in surplus is \(\Delta S\approx-D_AA\Delta y+D_LL\Delta y\), where \(D_A\) and \(D_L\) are asset and liability modified durations. Matching duration alone is not enough if asset and liability values differ: first-order surplus sensitivity is neutral when their dollar durations, \(D_AA\) and \(D_LL\), are equal.
Duration matching is an approximation for small, parallel changes in yields with fixed cash flows. It does not generally eliminate the effects of large changes, nonparallel yield-curve shifts, different convexities, or changes in the cash flows themselves. Those limitations matter when assessing a portfolio under stress.
Worked Example: Equal Values and Duration but Different Cash Flows
A hypothetical insurer has assets paying $50 million at year 1 and $55.125 million at year 3. It owes $105 million at year 2. Initially, the annual spot rate is 5% at all maturities. First verify the initial values and durations. Then revalue the cash flows at a parallel 10% rate. Report values and surplus in millions to three decimal places.
Method. Compare present values and their weighted payment times at the initial flat rate, then reprice each cash flow at the stressed rate. Matching duration can reduce first-order sensitivity, but it does not make different dated cash flows identical.
Initial values. Asset present values are \(50/1.05=47.619048\) million and \(55.125/(1.05)^3=55.125/1.157625=47.619048\) million. Thus \(A=95.238095\) million. The liability value is \(105/(1.05)^2=105/1.1025=95.238095\) million. Initial surplus is zero.
Initial durations. The asset payments have equal present values, so \(D_{\text{Mac},A}=(47.619048(1)+47.619048(3))/95.238095=2\) years. The single liability payment is at year 2, so \(D_{\text{Mac},L}=2\) years. Both values are equal, so their modified dollar durations are equal as well.
Values at 10%. Asset value is \(50/1.10+55.125/(1.10)^3=45.454545+41.416228=86.870774\) million. Liability value is \(105/(1.10)^2=105/1.21=86.776860\) million. Stressed surplus is \(86.870774-86.776860=0.093914\) million. Rounded to three decimals, assets are $86.871 million, liabilities are $86.777 million, and surplus is $0.094 million. Although initial values and durations matched, the different payment timing leaves a nonzero surplus after the large rate change.
Managing Surplus Alongside Cash Needs
Surplus management treats \(S=A-L\) as a risk measure to monitor under base-case valuations and market scenarios. An insurer can examine changes in surplus under parallel and nonparallel rate shifts, while also checking whether projected asset receipts cover liability payments. The selected scenarios should reflect the insurer’s exposures; a single parallel shift does not represent every possible movement in a yield curve.
The approaches can be compared as follows:
| Approach | Main question | What it can address | Important limitation |
|---|---|---|---|
| Cash flow matching | Will asset cash be available when payments are due? | Timing and amount of scheduled receipts relative to payments | Depends on cash flows occurring as expected and on suitable asset-liability matching |
| Duration matching | How sensitive are market values to a small parallel rate move? | First-order interest-rate sensitivity when dollar durations are aligned | Does not ensure payment-date liquidity or fully offset large or nonparallel shocks |
| Surplus management | How does the value of assets less liabilities change under scenarios? | Overall market-value balance and scenario-based assessment | A positive surplus alone does not demonstrate that cash is available when needed |
List expected asset receipts and liability payments by date, including material uncertainty in their timing or amount.
Use the relevant valuation basis and discount rates to calculate asset value, liability value, and surplus.
Assess surplus under market scenarios, then separately check whether liquid resources cover payments when due.
Common Exam Trap
A common error is to treat a positive market-value surplus as proof that every liability payment can be met from asset receipts available at the same date. In the second worked example, initial surplus is positive at $3.719 million, but the year-2 cash shortfall is $35.900 million if the insurer has no other funds. The incorrect conclusion, “surplus is positive, so there is no shortfall,” confuses present value with payment-date cash.
The correction is to answer both questions separately: value assets and liabilities at the valuation date to calculate surplus, and then compare cash available with cash due at each payment date. Similarly, do not use a duration match as a substitute for cash flow matching. Duration describes an approximate market-value response; it does not move a year-3 receipt forward to year 2.
Check Your Understanding
For each problem, distinguish market value, payment timing, and rate sensitivity as appropriate. All amounts are hypothetical.
- An insurer owes $60 million at year 2 and holds a fixed asset paying $60 million at year 2. Under a common spot curve, what is the present-value surplus from these two cash flows, and does changing the curve alter that matched contribution? (Answer: The contribution is $0 at any common spot curve, provided both payments occur as specified.)
- At a flat annual rate of 5%, assets pay $20 million at year 1 and $22.05 million at year 3; a liability pays $42 million at year 2. Calculate the asset and liability values and Macaulay durations. (Answer: Assets are worth $38.095 million, liabilities are worth $38.095 million, and both Macaulay durations are 2 years.)
- An insurer owes $70 million at year 2. Its only relevant asset receipt before then is $25 million at year 1, reinvested at 4% for one year; another asset pays at year 3. If there are no other funds, find the year-2 cash shortfall. (Answer: The year-1 receipt grows to $26 million, so the year-2 shortfall is $44 million.)
- Asset value is $120 million with modified duration 4, and liability value is $100 million with modified duration 3.5. Using the small parallel-shift approximation, estimate the change in surplus for a 1 percentage-point rise in rates. (Answer: \(-4(120)(0.01)+3.5(100)(0.01)=-$1.300\) million.)
- A portfolio has asset value $80 million and liability value $84 million, with both fixed cash flows paid at year 1. The annual rate rises from 5% to 6%. Calculate the initial and stressed surplus. (Answer: Initial surplus is \((80-84)/1.05=-$3.810\) million; stressed surplus is \((80-84)/1.06=-$3.774\) million, rounded to three decimals.)