Selective Lapses and Anti-Selection

Why the Composition of Lapses Matters

A lapse stress is not always just a change in the number of policies that remain in force. If policyholders with different risk characteristics lapse at different rates, the surviving portfolio can have a different risk profile from the original portfolio. For example, if lower-risk policyholders are more likely to lapse from a protection product, the remaining policyholders may have a higher average claim probability. This is a form of anti-selection: the change in who remains can make the insurer’s exposure less favorable.

The effect depends on the association between risk and lapse behavior. A higher lapse rate does not, by itself, imply that the remaining portfolio is riskier. The key calculation is the risk-weighted composition of the survivors. The methods here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Suppose a portfolio is divided into risk groups $j=1,\ldots,m$. Let $n_j$ be the initial exposure in group $j$, $q_j$ its probability of the event being studied over a specified period, and $l_j$ its probability of lapsing before that event period. If lapse and event are treated as ordered so that $l_j$ determines which exposure remains for the period, the expected surviving exposure in group $j$ is $n_j(1-l_j)$. The event probability for the surviving portfolio is $$ q_{\mathrm{surv}}= \frac{\sum_{j=1}^{m} n_j(1-l_j)q_j} {\sum_{j=1}^{m} n_j(1-l_j)}. $$ This is a weighted average of group-specific event probabilities, weighted by surviving exposure. It applies when group-specific $q_j$ values remain appropriate for the period being analyzed; if lapse and claims occur concurrently or depend dynamically on changing policy status, a model of their timing and dependence is needed instead.

Deriving the Survivor-Portfolio Formula

The formula follows by counting expected events and expected exposure within each group. Group $j$ contributes $n_j(1-l_j)$ surviving exposure units. At event probability $q_j$, its expected events are $n_j(1-l_j)q_j$. Summing these quantities gives total expected events; dividing by total surviving exposure gives the average event probability among survivors.

The same result can be viewed as conditioning. Let $R$ denote the risk group and $S$ the event that a policy survives the lapse selection. Then $P(\text{event}\mid S)=\sum_j P(\text{event}\mid R=j,S)P(R=j\mid S)$. When the event probability within group $j$ is $q_j$, and the surviving group weight is proportional to $n_j(1-l_j)$, this conditional probability is exactly the weighted formula above. The risk mix changes through the weights, even if the group-specific probabilities do not change.

Describe the risk groups.
State each group’s initial exposure and event probability for the same time period.

Apply group-specific lapse rates.
Multiply each initial exposure by $1-l_j$ to find its surviving exposure.

Recalculate expected events.
Multiply each surviving exposure by its group event probability and sum.

Normalize and interpret.
Divide expected events by total surviving exposure, then compare with the original portfolio rate.

Measuring the Change in Average Risk

The original event probability is $q_{\mathrm{initial}}=\sum_j n_jq_j/\sum_jn_j$. Comparing it with $q_{\mathrm{surv}}$ isolates the change in average risk per surviving exposure. The total expected number of events can still fall because fewer policies remain, even while the event probability per survivor rises. These are different measures and should not be confused.

Worked Example: Lower-Risk Policyholders Lapse More Often

A hypothetical protection portfolio has 40 high-risk policies with a one-year claim probability of 0.10 and 60 lower-risk policies with a probability of 0.02. Over the lapse period, 10% of the high-risk group and 50% of the lower-risk group lapse. Each claim costs $25,000. Calculate the initial and surviving claim probabilities and the expected claim cost per policy in each portfolio.

Method. Reweight the two groups by their surviving exposures, then calculate expected claims and divide by exposure. This avoids applying an overall lapse rate to a portfolio whose groups lapse differently.

Initially, expected claims are $40(0.10)=4.0$ in the high-risk group and $60(0.02)=1.2$ in the lower-risk group. Thus, there are $4.0+1.2=5.2$ expected claims among 100 policies, giving an initial claim probability of $5.2/100=0.052$.

After lapses, the high-risk group has $40(1-0.10)=36$ survivors. The lower-risk group has $60(1-0.50)=30$ survivors. Total surviving exposure is $36+30=66$.

Expected claims among survivors are $36(0.10)=3.6$ and $30(0.02)=0.6$, for a total of $4.2$. The survivor claim probability is $4.2/66=0.06363636\ldots$, or 0.0636 to four decimal places. It has risen from 0.0520, even though expected total claims fell from 5.2 to 4.2.

At $25,000 per claim, initial expected claim cost per policy is $0.052(25{,}000)=\$1{,}300.00$. For survivors it is $(4.2/66)(25{,}000)=\$1{,}590.91$, rounded to the nearest cent. The average cost has increased by $\$1{,}590.91-\$1{,}300.00=\$290.91$ per policy. As a check, the survivor group’s total expected claim cost is $4.2(\$25{,}000)=\$105{,}000$, and $\$105{,}000/66=\$1{,}590.91$ per survivor.

Common mistake avoided: Applying the overall lapse rate of 34% to both groups would ignore that lower-risk policies lapse more often. It would preserve the original risk mix and therefore miss the increase in average claim probability.

Use Exact Survivor Weights Before Rounding

Small rounding errors in a survivor probability can become material when multiplied by a large benefit. Keep expected counts and exposure weights unrounded through the cost calculation; round the final monetary result only.

Worked Example: Exact Claim Cost After Selective Lapses

A hypothetical portfolio starts with 65 high-risk exposure units, each with claim probability 0.08, and 55 lower-risk units, each with probability 0.02. After selective lapses, 44.375 high-risk units and 20.625 lower-risk units remain. The claim amount is $30,000. Find the initial and survivor claim costs per exposure unit, and the increase in the survivor average.

Method. Calculate expected claims from each group before dividing by total exposure. This preserves exactness when the resulting frequency has repeating decimals.

Initially, expected claims are $65(0.08)=5.2$ and $55(0.02)=1.1$. The initial total is $5.2+1.1=6.3$ expected claims among $65+55=120$ exposure units. Thus, the initial claim probability is $6.3/120=0.0525$, and the initial expected claim cost per unit is $0.0525(\$30{,}000)=\$1{,}575.00$.

Among survivors, expected claims are $44.375(0.08)=3.55$ and $20.625(0.02)=0.4125$. Total expected claims are $3.55+0.4125=3.9625$, and total surviving exposure is $44.375+20.625=65$. Therefore, the exact survivor probability is $3.9625/65=0.060961538\ldots$.

The survivor expected claim cost is $(3.9625/65)(\$30{,}000)=\$1{,}828.846153\ldots$, or $1,828.85 to the nearest cent. The increase over the initial cost is $\$1{,}828.846153\ldots-\$1{,}575.00=\$253.846153\ldots$, or $253.85 to the nearest cent.

Common mistake and correction: Rounding the survivor probability to 0.061 first gives $0.061(\$30{,}000)=\$1{,}830$, and an apparent increase of $255. That is not the cent-accurate result. Use the unrounded ratio $3.9625/65$ for the cost calculation, and round only at the final step.

As a check, the survivor expected claims times the benefit equal $3.9625(\$30{,}000)=\$118{,}875$. Dividing by 65 exposures gives $\$118{,}875/65=\$1{,}828.846153\ldots$, consistent with the calculation above.

Consequences for Reserves and Profit

A portfolio-level reserve is also a weighted result. If risk groups have different per-policy reserves, changing their proportions changes the average reserve per survivor. A reserve calculated using the original average may therefore be inappropriate for the selected portfolio. This does not mean that every adverse lapse pattern automatically creates an immediate loss: the lapse payment, reserves released, and value of the remaining liabilities all matter and must be measured on a consistent basis.

Worked Example: Reassessing Reserves After a Selective Lapse

A hypothetical portfolio has 40 high-risk policies, each with a reserve of $12,000, and 60 lower-risk policies, each with a reserve of $8,000. During a period, 10% of the high-risk group and 40% of the lower-risk group lapse. Each lapse receives a $2,000 surrender payment. Assume the stated reserves are appropriate per policy for the relevant valuation time, and ignore expenses, taxes, and other cash flows. Calculate the remaining reserve, total post-lapse liability including surrender payments, and the error from using the original average reserve per survivor.

Method. Calculate reserves by group before and after lapses, and keep surrender payments separate from the reserve on policies that remain in force.

Initially, the high-risk reserve total is $40(\$12{,}000)=\$480{,}000$, and the lower-risk reserve total is $60(\$8{,}000)=\$480{,}000$. The initial total reserve is $\$960{,}000$, or $\$960{,}000/100=\$9{,}600$ per policy on average.

The high-risk survivors number $40(0.90)=36$; the lower-risk survivors number $60(0.60)=36$. The reserve for survivors is $36(\$12{,}000)+36(\$8{,}000)=\$432{,}000+\$288{,}000=\$720{,}000$.

There are $40(0.10)=4$ high-risk lapses and $60(0.40)=24$ lower-risk lapses, or 28 lapses in total. Surrender payments are $28(\$2{,}000)=\$56{,}000$. Total post-lapse liability, including those payments and the remaining reserve, is $\$56{,}000+\$720{,}000=\$776{,}000$. Relative to the initial reserve, the decrease is $\$960{,}000-\$776{,}000=\$184{,}000$, under the stated simplifications.

If the original average reserve of $9,600 were incorrectly applied to all 72 survivors, the calculated reserve would be $72(\$9{,}600)=\$691{,}200$. That is $\$720{,}000-\$691{,}200=\$28{,}800$ below the group-specific survivor reserve. The correct survivor average is $\$720{,}000/72=\$10{,}000$.

Common mistake avoided: Counting only the reduction in policy count, or applying the original average reserve to survivors, ignores the changed group mix. The calculation checks by reconciling each liability component: $56,000 of surrender payments plus $720,000 of remaining reserves equals $776,000.

Interpretation and Exam Traps

Selective lapses matter because claim costs and reserves are averages over a particular portfolio mix. For each risk group, compare the group’s lapse rate with its risk level, then calculate the survivor weights. If higher-risk groups lapse more, the average risk among survivors can improve; if lower-risk groups lapse more, it can worsen. If lapse rates are equal across groups, the relative weights are unchanged and the mix alone does not alter the average risk, although the total number of policies and expected claims decline.

Do not infer direction from lapse volume alone. A large overall lapse rate can improve or worsen the remaining mix.
Separate frequency from total claims. Expected claims per survivor can rise while total expected claims fall.
Use consistent denominators. Divide survivor expected events by survivor exposure, not by initial exposure.
Keep risk-group assumptions visible. A single pooled lapse rate conceals selection when lapse behavior differs by risk group.
Do not overstate the conclusion. A higher average risk is not automatically a larger total loss or a reserve deficiency; the relevant exposure, benefit, reserve, and cash flows must also be evaluated.

Key takeaway: The effect of selective lapses is a reweighting problem. Calculate each group’s surviving exposure, then recompute the portfolio quantity of interest using those weights. Keep group-specific risks distinct from changes in the overall portfolio mix.

Check Your Understanding

For each problem, calculate using the stated exposure units and time period. Round probabilities to four decimal places and money to the nearest cent where applicable.

A hypothetical portfolio has 50 policies in group A with a one-year claim probability of 0.09 and 50 in group B with a probability of 0.03. Lapse rates are 20% for A and 40% for B. What is the survivor claim probability, and is it higher or lower than the initial probability? (Answer: survivor probability $4.5/70=0.0643$; it is higher than the initial 0.0600.)
A portfolio begins with 30 higher-risk units at claim probability 0.08 and 70 lower-risk units at 0.02. Lapse rates are 10% and 30%, respectively. Find the initial and survivor claim probabilities. (Answer: initial $3.8/100=0.0380$; survivor $3.14/76=0.0413$.)
A policy block has 25 high-risk survivor units at claim probability 0.12 and 40 lower-risk survivor units at 0.025. Each claim costs $18,000. Find expected claims and expected claim cost per survivor exposure unit. (Answer: expected claims $25(0.12)+40(0.025)=4.0$; cost per survivor $4.0(\$18{,}000)/65=\$1{,}107.69$.)
Initially, 20 policies have reserves of $15,000 each and 80 have reserves of $9,000 each. After selective lapses, 14 of the first group and 48 of the second remain. What is the remaining reserve, and how much would an estimate using the initial average reserve per survivor understate it? (Answer: remaining reserve $14(\$15{,}000)+48(\$9{,}000)=\$642{,}000$; initial average is $10,200, so the pooled estimate is $62(\$10{,}200)=\$632{,}400$, an understatement of $9,600.)
Two groups have event probabilities 0.04 and 0.10. Their lapse rates are 25% and 10%, respectively. Without calculating a numerical portfolio rate, state whether the survivor mix shifts toward higher or lower event risk, and explain. (Answer: toward higher event risk, because the lower-risk group lapses at the higher rate, leaving a greater relative weight of the higher-risk group.)

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