Why Decompose an NPV Change?
When a revised profit test produces a different net present value, the total difference does not explain its cause. Mortality, morbidity, expenses, interest, and the reserve basis can each affect projected profits, and their effects may interact. A useful analysis changes one assumption component at a time, recalculates the relevant projection, and records the change in NPV.
The calculation is only meaningful if the scenarios use the same cash-flow definitions, projection horizon, and timing conventions. For a multi-state product, a changed mortality or morbidity assumption can alter the number of policies in each state, which may change premiums, claims, expenses, and reserves together. Therefore, “change one assumption” means change that assumption and rerun the projection; it does not mean adjust only one line of the original profit signature.
NPV Requires Correct Cash-Flow Timing
Let \(C_t\) be an expected net insurer cash flow at time \(t\), positive for an inflow and negative for an outflow. At effective annual interest rate \(i\), with \(v=1/(1+i)\), the NPV is the sum of each cash flow discounted from its actual payment date:
A profit signature is a sequence of profits at stated times, often at the ends of projection years. If a test instead lists premiums and expenses at the start of a year and claims at the end, those items must be placed at their own dates before discounting. Combining them into an annual total and discounting that total at year-end changes the timing and generally changes the NPV.
The basic attribution identity follows by adding successive NPV differences. If \(N_0\) is the baseline NPV and \(N_j\) is the NPV after the \(j\)th assumption substitution, define the \(j\)th contribution as \(\Delta_j=N_j-N_{j-1}\). Then:
Each intermediate NPV appears once with a positive sign and once with a negative sign, so the terms cancel. This reconciliation is an arithmetic identity; it does not imply that the contribution assigned to each assumption is unique. If assumption changes interact, changing their order can change the individual contributions.
Worked Example: Morbidity Change With Start-of-Year and Year-End Cash Flows
A hypothetical block of disability income contracts is projected for two years. The effective annual interest rate is 5%. There is no lapse. The fixed annual transition assumptions are active-to-disabled probability 0.08, active-to-death probability 0.01, disabled-to-active probability 0.20, and disabled-to-death probability 0.03; these apply to each year’s transition projection. The listed dollar amounts are the aggregate expected cash flows produced by the multi-state projection under the stated assumptions. Premiums less expenses are paid at the start of each year. Claims are paid at the end of the year. In the baseline, start-of-year net inflows are $12,000 at time 0 and $8,400 at time 1; year-end claims are $1,680 at time 1 and $2,310 at time 2. A morbidity-only revision increases those claims to $2,100 and $2,835, respectively, while leaving all other assumptions and start-of-year inflows unchanged. Find the morbidity contribution to NPV.
Method. Put every item at its stated payment date. The time-1 cash flow includes both the start-of-year inflow and the end-of-year claim, but those payments occur at different times: the inflow is at time 1 and the claim is at time 2 only if the year-end schedule is interpreted from the start of the year. Here the problem states that the second-year start-of-year inflow is at time 1 and the first-year year-end claim is also at time 1, so those two amounts are combined at time 1.
The baseline cash flows are therefore $12,000 at time 0, \(8{,}400-1{,}680=\$6{,}720\) at time 1, and \(-\$2{,}310\) at time 2. Thus:
Under the revised morbidity assumption, the time-1 net cash flow is \(8{,}400-2{,}100=\$6{,}300\), and the time-2 claim is $2,835. The revised NPV is:
The morbidity contribution is \(15{,}428.571429-16{,}304.761905=-\$876.19\), rounded to the nearest cent. The check is direct: the claim increases are $420 at time 1 and $525 at time 2, so their present value is \(420/1.05+525/1.1025=\$876.19\). The NPV reduction equals that present value.
Common mistake avoided: discounting a whole annual profit as though every payment occurred at year-end. The start-of-year inflow and the year-end claim do not share a payment date; placing them correctly is essential to the result.
Sequential Attribution Across Several Components
The following hypothetical example uses end-of-year profit signatures in thousands of dollars. Each signature value is already a projected net profit at the end of the stated year, including the profit-test effect of the applicable reserves. The projection has three dates: time 0, year 1, and year 2. The initial time-0 entry is included without discounting.
Worked Example: A Sequential Multi-State Profit-Test Bridge
A hypothetical long-term care block has active, disabled, and dead states, with no lapses. The baseline annual transition probabilities are active-to-disabled 0.10, active-to-dead 0.02, disabled-to-active 0.15, and disabled-to-dead 0.04. A projection team supplies the profit signatures below, in thousands, after applying each scenario’s assumptions. The baseline effective annual interest rate is 4%. Starting from the baseline, change mortality, then morbidity, then expenses, then the interest rate to 5%, and finally the reserve basis. Find each sequential contribution and reconcile to the final NPV change.
| Scenario after each step | Profit signature (time 0, year 1, year 2), $000 |
|---|---|
| Baseline | (2, 8, 10) |
| Mortality revised | (1, 7, 9) |
| Morbidity also revised | (0, 6, 8) |
| Expenses also revised | (-1, 5, 7) |
| Interest changed to 5% | (-1, 5, 7) |
| Reserve basis also revised | (-2, 5, 8) |
Step 1: Baseline NPV. At 4%, \(v=1/1.04\), and \(v^2=1/1.0816\). Therefore:
Step 2: Mortality contribution. Recalculate the signature with mortality changed and all other components held at baseline:
Thus \(\Delta_{\mathrm{mort}}=N_1-N_0=-2.886095\) thousand.
Step 3: Morbidity contribution. With the mortality change retained, revise morbidity and rerun the projection:
Therefore \(\Delta_{\mathrm{morb}}=N_2-N_1=-2.886095\) thousand, with rounding applied only to the displayed result.
Step 4: Expense contribution. At the revised mortality and morbidity assumptions, the expense change gives:
Hence \(\Delta_{\mathrm{exp}}=N_3-N_2=-2.886095\) thousand.
Step 5: Interest contribution. Keep the signature fixed and use 5%, so \(v=1/1.05\) and \(v^2=1/1.1025\):
The interest contribution is \(\Delta_i=N_4-N_3=-0.168475\) thousand.
Step 6: Reserve-basis contribution and reconciliation. Under the revised reserve basis, the signature is \((-2,5,8)\), still discounted at 5%:
Thus \(\Delta_{\mathrm{reserve}}=N_5-N_4=-0.092971\) thousand. The total change is \(10.018141-18.937870=-8.919729\) thousand. Adding the sequential contributions gives \(-2.886095-2.886095-2.886095-0.168475-0.092971=-8.919731\) thousand at the displayed precision; the small difference of $0.002 thousand is from rounding each displayed contribution. Using unrounded values, the contributions sum exactly to \(N_5-N_0\).
The reserve-basis change moves one thousand dollars of profit from time 0 to year 2: the signature changes by \((-1,0,+1)\). Its contribution is therefore \(-1+1/1.1025=-0.092971\) thousand. This is a timing effect in the profit signature, not an additional change to the interest rate.
Common mistake avoided: calculating every contribution against the original baseline and then adding the results. That approach generally double counts interactions. In a sequential bridge, each contribution compares adjacent scenarios, so the contributions reconcile to the endpoint change.
Why the Attribution Order Can Matter
The sequential method assigns each interaction to the assumption change that occurs later in the chosen sequence. A different order can give different individual contributions even though the total difference between the same endpoints cannot change. The following example isolates this point.
Worked Example: Mortality and Expense Interaction
A hypothetical two-year multi-state product has a baseline end-of-period profit signature \((0,10)\), in thousands, at an effective annual interest rate of 4%. The mortality-only scenario gives \((-1,8)\), the expense-only scenario gives \((-2,9)\), and the scenario with both changes gives \((-4,8)\). These are complete reprojected signatures; all other assumptions are held fixed. Compare the mortality and expense contributions under the two possible orders.
The baseline NPV is \(0+10/1.04=9.615385\) thousand. With mortality changed first, the NPV is:
The both-changed NPV is \(-4+8/1.04=3.692308\) thousand. Changing expenses after mortality contributes \(3.692308-6.692308=-3.000000\) thousand. The sequential contributions sum to \(-5.923077\) thousand.
Now change expenses first. Its NPV is \(-2+9/1.04=6.653846\) thousand, so \(\Delta_{\mathrm{exp,first}}=6.653846-9.615385=-2.961538\) thousand. Changing mortality after expenses contributes \(3.692308-6.653846=-2.961538\) thousand. These contributions also sum to \(-5.923077\) thousand, the same total change.
If an order-neutral split is required for these two assumption changes, average each component’s contribution across the two orders. Mortality receives \((-2.923077-2.961538)/2=-2.942308\) thousand, and expenses receive \((-3.000000-2.961538)/2=-2.980769\) thousand. Their sum remains \(-5.923077\) thousand. This averaging is a reporting convention, not a claim that the underlying interaction has a single uniquely determined cause.
Common mistake avoided: reporting one sequential contribution as though it were independent of ordering. State the order used, or specify the averaging convention, whenever the separate source effects matter.
Practical Checks Before Reporting
An attribution is easiest to audit when each scenario is defined precisely. In particular, record whether the revised assumption is an experience assumption used to project expected cash flows or a valuation assumption used to determine reserves. Changing both at once prevents a clean source attribution. A reserve-basis revision should be identified as such, with the resulting profit signature recalculated under that basis.
Before accepting a bridge, check that the endpoint scenarios match the requested baseline and revised cases, that all cash flows are discounted from their actual dates, and that every intermediate scenario changes only the intended component. Keep unrounded NPVs internally; round only the final reported amounts. Finally, verify that the signed contributions sum to the total NPV change.
Check Your Understanding
Use the stated timing and discount rate in each problem. Treat all amounts as hypothetical. Report dollar answers to the nearest cent unless the problem states thousands.
- A profit signature is \((3,6)\) thousand at times 0 and 1, with \(i=5\%\). A mortality-only revision produces \((2,5)\) thousand. Find the NPV contribution. (Answer: \(-1-1/1.05=-1.952381\) thousand, or \(-\$1,952.38\))
- At \(i=4\%\), a revised reserve basis changes a three-date signature by \((0,-2,+3)\) thousand at times 0, 1, and 2. Find the NPV effect. (Answer: \(-2/1.04+3/1.0816=+0.8505917\) thousand, or \(+\$850.59\))
- At \(i=4\%\), the baseline signature is \((1,8)\) thousand and the mortality-only signature is \((0,7)\) thousand. An expense change after mortality produces \((-1,6)\) thousand. Find the mortality-first contribution, the subsequent expense contribution, and the total change. (Answer: mortality \(-1-1/1.04=-1.961538\) thousand; expense \(-1-1/1.04=-1.961538\) thousand; total \(-3.923077\) thousand)
- A hypothetical product has a baseline signature \((0,12)\) thousand and revised signature \((-2,10)\) thousand, with \(i=5\%\). The revised case includes both a morbidity and an expense change. Find the total NPV change and explain whether the total alone identifies each source contribution. (Answer: \(-2-2/1.05=-3.904762\) thousand, or \(-\$3,904.76\); no, separate contributions require intermediate projections or an allocation convention)