Introduction
Lapse experience is not automatically favorable or unfavorable to an insurer. A terminating policy may release a reserve larger than its surrender payment, producing an immediate gain. But the same termination also removes future premiums, and may eliminate profitable future margins. Its effect depends on the policy’s reserve, surrender terms, expenses, and expected future cash flows.
For participating business, lapse and surrender experience is one source of the difference between expected and actual surplus. The useful question is not simply whether more or fewer policies terminated than expected. It is what financial effect each additional or missing termination has, and whether the comparison includes only the current period or also future policy value.
Immediate Surplus Effect of a Termination
Consider a hypothetical policy that terminates at a valuation point. Let \(R\) be the reserve held immediately before termination, \(S\) the surrender payment, and \(E_s\) any immediate expense caused by processing the termination. If the insurer releases the reserve when the policy ends, the immediate change in surplus attributable to the termination is:
A positive \(g\) is an immediate gain per termination; a negative \(g\) is an immediate loss. This calculation assumes that the stated reserve is released, the surrender payment and termination expense are paid at the same point in the accounting comparison, and no other cash flows change in this calculation. If a contract has no surrender payment, set \(S=0\). If termination expenses vary by policy, use the expense applicable to that policy.
For \(n\) policies exposed during a period, suppose the expected lapse rate is \(q_e\), the actual lapse rate is \(q_a\), and the same immediate gain \(g\) applies to every termination. Expected terminations are \(nq_e\), while actual terminations are \(nq_a\). The actual-minus-expected immediate lapse variance is therefore:
This is a comparison of realized experience with the chosen expected basis; it is not a statement that the realized number of terminations must equal a fractional expected count. If actual counts are supplied directly, use \(N_a-N_e\) in place of \(n(q_a-q_e)\).
Why the Lapse-Variance Formula Holds
The formula follows by valuing the termination event in each experience basis. Under the expected basis, \(nq_e\) terminations produce immediate surplus of \(nq_eg\). Under actual experience, \(nq_a\) terminations produce immediate surplus of \(nq_ag\). Subtracting expected from actual gives:
The derivation requires the same per-termination gain \(g\) to apply to the policies being compared. If reserves, surrender payments, or expenses differ materially by duration, product, or policyholder group, calculate the variance separately for each group and then add the results. Applying one average \(g\) to a changing mix of policies can give the wrong sign or magnitude.
Worked Example: More Lapses Than Expected
A hypothetical block has 1,000 policies exposed for one year. The expected lapse rate is 5%, but 80 policies actually lapse. At termination, each policy has a reserve of $12,000, a surrender payment of $10,800, and a termination expense of $100. Calculate the immediate lapse-experience variance relative to expectation.
Method. First calculate the immediate gain per termination, then compare actual and expected lapse counts. This avoids treating the whole surrender payment as a loss without recognizing the reserve release.
Expected terminations are \(1{,}000(0.05)=50\). Actual terminations are 80, so the excess is \(80-50=30\).
The immediate gain per termination is:
Expected immediate surplus from terminations is \(50(\$1{,}100)=\$55{,}000\). Actual immediate surplus is \(80(\$1{,}100)=\$88{,}000\). Thus the variance is:
The block has a $33,000 immediate gain relative to expected lapse experience. The check is that the excess count is 30, not 80: the first 50 actual terminations are already represented in the expected basis.
Terminations Also Change Other Cash Flows
The reserve-release calculation isolates the immediate termination entry. A complete period comparison may also need to reflect premiums and expenses that differ because policies terminate at different times. For example, if a policy lapses at the start of the period, it may pay no premium for the period and incur no per-policy maintenance expense. If it remains in force, those cash flows may occur. The assumed timing is essential: a policy that lapses near year-end may have paid most of the period’s premium and incurred most of its expenses.
Termination also removes future coverage, future claims, and future premium income. These effects are not automatically captured by \(R-S-E_s\) as a measure of all future economic value. Keep the accounting comparison and any separate valuation of future margins conceptually distinct. In particular, do not add a value of future profits to a surplus calculation unless the definitions establish that the amounts do not overlap.
| Change associated with termination | Possible effect on insurer value | What determines the direction? |
|---|---|---|
| Reserve released and surrender paid | Immediate gain or loss | Reserve compared with surrender payment and termination expense |
| Future premiums no longer received | May reduce value | Premium amounts, timing, and the remaining policy term |
| Future expenses avoided | May increase value | Expenses that would have been incurred while the policy remained in force |
| Future benefits no longer payable | May increase value | Benefit amounts, incidence, and the policy’s remaining coverage |
| Future net margins lost | May reduce value if margins were positive | The premiums, expenses, and benefits expected on continuing coverage |
Worked Example: Fewer Lapses and Period Cash Flows
A hypothetical block has 500 policies exposed at the start of a year. Expected lapses are 12%, but actual lapses are 45 policies. Each termination releases a reserve of $15,000, requires a surrender payment of $14,000, and incurs a $200 termination expense. Assume lapses occur at the start of the year. Each policy that remains in force for the full year produces $1,000 of premium, $300 of maintenance expense, and $500 of expected benefit cost. Calculate the combined period variance from termination effects and these current-year cash flows.
Method. Compare actual and expected termination gains, then separately compare the number of policies continuing for the year. This avoids omitting the cash flows on the additional policies that remain in force.
Expected lapses are \(500(0.12)=60\), so actual lapses are 15 fewer than expected. The immediate gain per termination is:
Expected immediate termination surplus is \(60(\$800)=\$48{,}000\); actual immediate termination surplus is \(45(\$800)=\$36{,}000\). The termination variance is therefore \(\$36{,}000-\$48{,}000=-\$12{,}000\).
Expected continuing policies are \(500-60=440\). Actual continuing policies are \(500-45=455\), or 15 more. The net current-year margin per continuing policy under the stated assumptions is:
The additional current-year margin is \(15(\$200)=\$3{,}000\). Combining only the two specified period effects gives:
The result is a $9,000 loss relative to the expected basis for these items. The check is that fewer lapses reduce the immediate gain by $12,000, while the extra policies contribute $3,000 of current-year margin.
Use Groups When Lapse Economics Differ
Lapse rates and per-policy effects may vary by duration, product, or other grouping. A block-level rate can conceal offsetting experience: one group may have more lapses than expected where each lapse creates an immediate gain, while another may have fewer lapses where each lapse would have created an immediate loss. Calculate the variance using the exposure and expected rate for each group, together with that group’s own termination economics.
Worked Example: Offsetting Experience Across Two Groups
A hypothetical insurer compares two groups over one year. Group A has 200 policies exposed, an expected lapse rate of 4%, and an actual lapse rate of 7%. For each termination, its reserve is $6,000, surrender payment is $5,500, and termination expense is $100. Group B has 100 policies exposed, an expected lapse rate of 10%, and an actual lapse rate of 6%. For each termination, its reserve is $11,000, surrender payment is $10,200, and termination expense is $100. Calculate the total immediate lapse-experience variance.
Method. Calculate the expected and actual count difference and the per-termination gain within each group. This avoids applying a pooled average termination gain to unlike policies.
For Group A, expected lapses are \(200(0.04)=8\) and actual lapses are \(200(0.07)=14\), an excess of 6. Its gain per termination is \(\$6{,}000-\$5{,}500-\$100=\$400\). Group A’s variance is \(6(\$400)=\$2{,}400\).
For Group B, expected lapses are \(100(0.10)=10\) and actual lapses are \(100(0.06)=6\), or 4 fewer than expected. Its gain per termination is \(\$11{,}000-\$10{,}200-\$100=\$700\). Group B’s variance is \((-4)(\$700)=-\$2{,}800\).
The total immediate variance is:
Thus the combined block has a $400 immediate loss relative to expectation, despite having more lapses than expected in Group A. The calculation checks by adding the two group variances, not by relying on the direction of either group’s rate alone.
Common Mistakes and Practical Checks
- Calling every increase in lapses a gain. More lapses increase immediate surplus only when the per-termination gain \(R-S-E_s\) is positive. If it is negative, more lapses increase the immediate loss.
- Using the actual count instead of the excess count. When measuring a variance against expectation, compare actual terminations with expected terminations. Do not count all actual terminations as unexpected.
- Reversing the sign for fewer lapses. If \(g>0\), fewer lapses mean fewer immediate gains and a negative variance. If \(g<0\), fewer lapses avoid some losses and produce a positive variance.
- Ignoring timing. The premium and expense effect depends on when policies terminate and which cash flows would otherwise have occurred.
- Combining unlike groups too early. Different reserves, surrender amounts, expenses, or lapse rates call for group-specific calculations before aggregation.
- Mixing immediate surplus with future value. The reserve release and surrender payment describe an immediate accounting effect under the stated assumptions. Lost future premiums and benefits require consistent treatment in any broader valuation.
Lapse experience matters because policy termination changes both the insurer’s immediate liability and the future stream of policy cash flows. A sound analysis states the exposure period, compares actual with expected terminations, calculates the effect per termination, and includes other cash-flow differences only on a clearly defined basis. This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, calculate the requested variance as actual experience minus expected experience. Unless a problem says otherwise, include only the specified immediate termination effects.
- A block has 800 policies exposed, an expected lapse rate of 6%, and an actual lapse rate of 8%. Per termination, the reserve is $7,200, the surrender payment is $6,900, and the expense is $50. What is the immediate variance? (Answer: $4,000 gain)
- A block of 300 policies has expected lapses of 15% and actual lapses of 36. The reserve per termination is $9,000, the surrender payment is $9,250, and the termination expense is $100. What is the immediate variance? (Answer: $3,150 gain)
- For 600 exposed policies, expected lapses are 10% and actual lapses are 13%. Each termination releases a $5,000 reserve, requires a $4,700 surrender payment, and incurs $100 expense. Calculate the immediate lapse variance. (Answer: $3,600 gain)
- Group A has 150 exposed policies, expected and actual lapse rates of 4% and 8%, and per-termination reserve, surrender payment, and expense of $10,000, $9,600, and $100. Group B has 250 exposed policies, rates of 8% and 6%, and corresponding amounts of $7,000, $7,250, and $50. Calculate the total immediate variance. (Answer: $3,300 gain)
- A group has 400 policies exposed, expected lapses of 10%, and actual lapses of 7%. Each termination releases a reserve of $14,000, requires a $13,200 surrender payment, and incurs $100 expense. What is the immediate variance, and why is its sign not the same as the per-termination gain? (Answer: $8,400 loss; actual lapses are 12 fewer than expected, so 12 expected gains of $700 each are absent.)