Separating Assumption Changes from Experience Deviations

Why the Distinction Matters

A valuation can change because the future is now expected to unfold differently, or because the past period did not unfold as expected. These are not the same event. A higher assumed future expense is a prospective assumption change; expenses already incurred above expectation are an experience deviation. Both may reduce surplus, but they enter the analysis through different parts of the accounting and valuation bridge.

Keeping the effects separate makes a result explainable. It also prevents a common error: comparing reserves on different bases and attributing the entire difference to experience. The approach below is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: An experience deviation is the difference between actual events or cash flows over a period and the corresponding events or cash flows expected under the assumptions in force for that period. An assumption change is a prospective change to the assumptions used to value future cash flows. A reserve is the present value of future net outgo under a stated valuation basis. With assets held fixed, an increase in a liability reserve reduces surplus by the same amount.

For clarity, use a positive reserve for a liability and define surplus as assets minus reserves. Thus, if a reserve changes from $V_{\text{old}}$ to $V_{\text{new}}$, with no simultaneous asset change, the surplus effect of remeasurement is

\Delta \text{surplus}_{\text{remeasurement}} = -(V_{\text{new}}-V_{\text{old}}).

A reserve increase is a loss to surplus; a reserve decrease is a gain. This statement concerns the valuation effect only. Actual premiums, claims, expenses, investment income, and other cash flows may also change assets and surplus during the period.

A Consistent Separation

A useful bridge has three stages. First value the expected in-force business using the original assumptions. Next replace expected period experience with actual experience while retaining those assumptions. Finally, revalue the actual in-force business using the revised assumptions. This order separates the experience effect from the prospective remeasurement effect.

Set the original basis.
Record the assumptions used to form the period expectation and the opening reserve. Specify which cash flows and in-force policies are included.

Measure actual experience on that basis.
Compare actual period cash flows and the actual in-force group with the expected outcomes. Do not silently substitute revised assumptions in this comparison.

Remeasure prospectively.
For the actual closing in-force group, calculate the reserve using the revised future assumptions. Compare it with the reserve for that same group on the original basis.

This staged method gives an exact decomposition when all comparisons use the same cash-flow definition and valuation date. It does not mean the real world changes in three separate steps. It is an attribution convention: a different ordering can allocate interaction effects differently, so the ordering should be stated.

Key result: The change from an expected closing reserve on the original basis to an actual closing reserve on the revised basis can be written as

V_{\text{actual,new}}-V_{\text{expected,old}} = \bigl(V_{\text{actual,old}}-V_{\text{expected,old}}\bigr) + \bigl(V_{\text{actual,new}}-V_{\text{actual,old}}\bigr).

The first bracket is the reserve effect of actual experience, evaluated on the old basis. The second bracket is the assumption-change effect, evaluated for the actual closing group. Under the stated surplus convention, negate each reserve change to obtain its corresponding surplus effect.

Why the Decomposition Holds

The result is an accounting identity, not an approximation. Add and subtract the reserve for the actual closing group valued on the old basis:

\begin{aligned} V_{\text{actual,new}}-V_{\text{expected,old}} &=V_{\text{actual,new}}-V_{\text{actual,old}}\\ &\quad+V_{\text{actual,old}}-V_{\text{expected,old}}. \end{aligned}

Rearranging the terms gives the stated bridge. Its usefulness depends on interpreting each comparison consistently. For example, the first bracket can reflect a different number of policies remaining in force, while the second changes the projected future cash flows per actual policy. If actual period claims or expenses have already been included in a profit calculation, do not count them again as a reserve change.

For a cash-flow comparison, define net outgo as benefits and expenses less premiums. Holding the original expectation fixed, actual net outgo above expected net outgo is adverse experience. If the cash flows occur at different times, compare them at the same measurement date, allowing for the applicable accumulation or discounting. A reserve remeasurement is not itself an actual cash payment.

Worked Example: A Change in the Discount Rate

A hypothetical insurer has a guaranteed liability of $1,500 payable exactly three years from the valuation date. There is no mortality contingency, no other cash flow, and no change in the amount or timing of the payment. The effective annual interest assumption changes prospectively from 4% to 5%. Find both reserves and the immediate surplus effect.

Method. Discount the same certain payment under each valuation basis. Because the payment is fixed and has no mortality contingency, only the discount rate changes.

At 4%, $v=1/1.04$, so the old reserve is

V_{\text{old}}=\frac{1500}{(1.04)^3} =\frac{1500}{1.124864} =1333.4945\ldots

At 5%, the new reserve is

V_{\text{new}}=\frac{1500}{(1.05)^3} =\frac{1500}{1.157625} =1295.7564\ldots

The reserve change is $1295.7564\ldots-1333.4945\ldots=-37.7381\ldots$. The reserve therefore decreases by $37.74 to the nearest cent. Since assets are unchanged at the instant of remeasurement, surplus increases by $37.74.

Check. The payment amount and term are unchanged, and the higher discount rate gives a lower present value. The direction and amount are consistent. The common mistake is to calculate or report a different decrease without subtracting these two present values; the calculation above gives $37.74, not $39.97.

Separating Experience from a New Mortality Basis

A change in the number of policies in force is experience, while a change in the mortality assumption for future periods is prospective remeasurement. They can occur together. To isolate them, first compare expected and actual in-force counts using the original mortality basis, then compare old and new mortality assumptions for the actual count.

Worked Example: Fewer Policies in Force and Higher Future Mortality

A hypothetical group has 10 identical policies at the start of a period. No claims, premiums, or expenses occur during the period. One lapse was expected, but two policies lapse, leaving 8 policies in force rather than the expected 9. Each remaining policy pays $1,000 at the end of the next year if death occurs during that year. The original one-year death probability is $q=0.02$; a prospective review changes it to $q=0.03$. The effective annual interest rate is 5%. Lapses have no surrender payment. Calculate the experience and assumption effects on the closing reserve and surplus.

Method. Value expected and actual in-force counts first on the original mortality basis, then revalue the actual count on the new basis. For $n$ policies, death probability $q$, benefit $B$, and annual effective interest rate $i$, the reserve is $nqB/(1+i)$.

Expected closing reserve on the original basis:

V_{\text{expected,old}} =\frac{9(0.02)(1000)}{1.05} =\frac{180}{1.05} =171.4286\ldots

Actual closing reserve on the original basis:

V_{\text{actual,old}} =\frac{8(0.02)(1000)}{1.05} =\frac{160}{1.05} =152.3810\ldots

The experience-related reserve change is $152.3810\ldots-171.4286\ldots=-19.0476\ldots$. The lower in-force count reduces the reserve by $19.05, which increases surplus by $19.05.

Now value the same 8 actual policies using the revised death probability:

V_{\text{actual,new}} =\frac{8(0.03)(1000)}{1.05} =\frac{240}{1.05} =228.5714\ldots

The assumption-change reserve effect is $228.5714\ldots-152.3810\ldots=76.1905\ldots$. The reserve increases by $76.19, reducing surplus by $76.19.

Combined check. The total reserve change from expected-old to actual-new is $228.5714\ldots-171.4286\ldots=57.1429\ldots$, an increase of $57.14. The two components also sum to $-19.0476\ldots+76.1905\ldots=57.1429\ldots$. Thus the net surplus effect is a decrease of $57.14. This example includes no current-period claims; if claims had occurred, their cash-flow experience would need to be included separately.

Common mistake avoided. Comparing the expected-old reserve directly with the actual-new reserve and labelling all $57.14 as a mortality assumption loss ignores the favorable effect of the lower in-force count. The staged comparison identifies both components.

Experience Profit and a Future Expense Update

A period’s actual expense deviation affects profit through cash outgo. A revised estimate of future expenses affects the closing reserve. Even when both arise from expense information, they should be recorded in their own parts of the bridge.

Worked Example: Actual Maintenance Costs and Future Expense Assumptions

A hypothetical block has 20 policies in force throughout a one-year period. The expected maintenance expense is $40 per policy, but actual expenses are $45 per policy, paid at the end of the year. There are no other cash flows. At the closing valuation date, a further expense is expected one year later for each policy. The old assumption is $30 per policy and the revised assumption is $35 per policy. Use an effective annual interest rate of 5%. Find the current-period experience profit effect and the closing reserve remeasurement effect.

Method. Compare actual and expected current-period outgo for experience profit. Separately, discount the future expense on each valuation basis for the 20 policies still in force.

Expected current-period expense is $20(40)=\$800$. Actual current-period expense is $20(45)=\$900$. Actual outgo exceeds expected outgo by $\$900-\$800=\$100$, so the experience profit effect is an adverse $100.

The old closing reserve for the future expense is

V_{\text{old}}=\frac{20(30)}{1.05} =\frac{600}{1.05} =571.4286\ldots

The new closing reserve is

V_{\text{new}}=\frac{20(35)}{1.05} =\frac{700}{1.05} =666.6667\ldots

The reserve increase is $666.6667\ldots-571.4286\ldots=\$95.2381\ldots$, or $95.24 to the nearest cent. It causes a $95.24 reduction in surplus. The $100 experience loss and $95.24 reserve remeasurement loss are distinct; their combined effect is an adverse $195.24 under the stated assumptions and timing.

Check. The $100 compares expenses already incurred with the period expectation. The $95.24 compares future liabilities for the same 20 policies. Neither amount is included twice.

Common Traps and Practical Checks

Changing the basis before measuring experience. Compare actual period outcomes with the expectation formed under the original period assumptions. Then show any new prospective basis separately.
Mixing the expected and actual in-force groups. To isolate a basis change, value the same actual closing group under both bases.
Reversing the surplus sign. With surplus equal to assets minus reserves, a $1 increase in reserve reduces surplus by $1, all else equal.
Counting a cash flow twice. A claim or expense paid during the period belongs in the period cash-flow result. A reserve is for future net outgo at the valuation date.
Ignoring timing. Compare amounts at a common date. A year-end deviation and a valuation-date reserve change cannot be added without considering the stated timing and accumulation convention.
Failing to state attribution order. When experience and assumptions interact, use a clear sequence. The bridge above measures experience first on the old basis and then revalues the actual closing group.

For a solvency or profit analysis, report the components rather than only the net. A compact explanation should identify the original expectation, actual period result, closing in-force group, reserve on each basis, and resulting surplus effects. This makes it possible to tell whether a loss arose from realized cash flows, the future outlook, or both.

Check Your Understanding

For each problem, keep actual experience separate from any prospective reserve remeasurement. Unless stated otherwise, a reserve increase reduces surplus dollar for dollar.

A certain $1,050 payment is due in one year. The effective annual interest assumption changes from 4% to 5%, with no other change. Find the reserve decrease and the immediate surplus effect, to the nearest cent. (Answer: Reserve decreases by $9.62; surplus increases by $9.62.)
A block of 12 policies was expected to incur $70 per policy of current-period expense. Actual expense is $74 per policy. What is the current-period experience profit effect, before any future reserve remeasurement? (Answer: Adverse experience of $48.)
For a future one-year death benefit, 15 policies were expected to remain in force, but 12 actually remain. The benefit is $2,000 per policy, the old death probability is 0.04, the revised probability is 0.05, and the effective annual interest rate is 4%. Find the experience and assumption reserve changes using the staged method. (Answer: Experience reduces reserve by $230.77; the assumption update increases reserve by $230.77; net reserve change is $0.00.)
Assets are $5,000 at a valuation date. The reserve is $4,200 on the old basis and $4,500 on the revised basis for the same actual in-force group. Separately, actual current-period outgo exceeded expected outgo by $60. State the remeasurement effect and the experience effect, without combining their timing. (Answer: Remeasurement reduces surplus by $300; current-period experience reduces profit by $60.)

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