Universal Life Profit Emergence Under Changing Persistency

Why Persistency Changes Profit Emergence

Universal life contracts combine an account value with insurance protection. Premium loads and policy charges can generate income for the insurer, while the account value grows according to the credited rate and charges. Death and lapse both end a policy, but they have different cash-flow consequences. A death can produce a net insurance claim; a lapse can produce a surrender payment and release the associated reserve.

A persistency comparison therefore requires two linked projections. First, project the account value for a policy that remains in force. Second, weight each duration’s profit by the expected number of policies still in force. Under the assumptions below, changing lapse rates changes the cohort weights and decrement-related cash flows, but not the account path for an individual survivor.

Model definition: At the start of year $t$, an in-force policy has account value $B_t$. It pays annual premium $P$, of which a proportion $\alpha$ is retained as a premium load. The insurer deducts account charges $e$ and $c$ at year-end; the account earns effective annual interest $i$. The end-year account value before death or lapse is $$ A_t=\bigl(B_t+(1-\alpha)P\bigr)(1+i)-e-c. $$

Assume the death benefit is a fixed amount $F$, with $A_t<F$, and the account value is applied against that benefit. The insurer’s net death claim is then $F-A_t$. On lapse, the policyholder receives $A_t-s$, where $s$ is the surrender charge retained by the insurer. Let $q_t$ be the probability of death during the year, $w_t$ the probability of lapse during the year, and $p_t=1-q_t-w_t$ the probability of surviving and persisting to the next year. Death and lapse are mutually exclusive decrements during each year.

Let $R_t$ be the insurer’s reserve at the start of year $t$ for the insurance risk, separate from the account-value liability. Its values are supplied on the chosen valuation basis. Let $E_t$ be the year-end expense per policy in force at the start of the year. This simplified annual model assumes no taxes, reinsurance, investment spread on account assets, or other cash flows. Premium-load income is accumulated to year-end at $i$; other stated charges, expenses, claims, surrender charges, and reserve cash flows are measured at year-end.

Annual profit formula: Under these assumptions, expected profit for the year per policy in force at the start of the year is $$ g_t=\alpha P(1+i)+e+c-E_t+(1+i)R_t-q_t(F-A_t)-w_t s-p_tR_{t+1}. $$ The expected profit for the whole cohort in year $t$ is $n_tg_t$, where $n_t$ is the expected number of policies in force at the start of that year.

Why the Profit Formula Works

The formula is a year-end cash-flow balance. The premium load and account charges are insurer income; expenses and expected net death claims are outgo. A lapse generates the retained surrender charge $s$. The opening risk reserve, accumulated at the valuation interest rate, is available to meet the year’s claims and expenses. At year-end, the expected closing reserve is needed only for policies that survive and persist, giving the deduction $p_tR_{t+1}$.

The weighting follows from the three mutually exclusive outcomes. Out of one policy in force at the start of the year, expected deaths are $q_t$, lapses are $w_t$, and survivors are $p_t$. Thus the expected closing reserve is $q_t(0)+w_t(0)+p_tR_{t+1}$, because the risk reserve is released when a policy exits. This gives the reserve movement $(1+i)R_t-p_tR_{t+1}$. Similarly, expected net death claims are $q_t(F-A_t)$, and expected surrender-charge income is $w_ts$. Combining these terms with the other year-end cash flows gives $g_t$.

Project the account.
Use the contract’s premium, load, charges, and crediting rate to obtain the end-year account value before decrements.

Calculate per-policy profit.
Apply mortality and lapse probabilities to the net claim, surrender charge, and closing reserve.

Project the cohort.
Use $n_{t+1}=n_tp_t$, then multiply each year’s per-policy profit by its starting cohort size.

Compare on a consistent basis.
Discount end-year profits to the issue date if comparing present values, and distinguish per-policy from aggregate results.

Worked Account and First-Year Profit

Worked Example: One Year of Account and Profit Emergence

Consider a hypothetical universal life contract with $B_0=\$5{,}000$, annual premium $P=\$1{,}000$, premium load $\alpha=0.05$, effective annual interest $i=0.05$, year-end account charges $e=\$40$ and $c=\$100$, and expense $E_0=\$50$. The fixed death benefit is $F=\$10{,}000$. For year 1, use $q_0=0.02$, lapse probability $w_0=0.10$, surrender charge $s=\$40$, and reserves $R_0=\$80$, $R_1=\$100$. Find the account value and expected profit per starting policy.

Method: project the account first, then apply the decrement and reserve terms. The amount credited to the account at the start of the year is the premium net of its load: $$ (1-\alpha)P=(1-0.05)(1000)=\$950. $$

The account earns 5% before the year-end charges are deducted: $$ A_0=(5000+950)(1.05)-40-100 =6247.50-140 =\$6{,}107.50. $$ The net death claim is $F-A_0=10{,}000-6{,}107.50=\$3{,}892.50$. The persistence probability is $$ p_0=1-q_0-w_0=1-0.02-0.10=0.88. $$

The premium-load income accumulated to year-end is $0.05(1000)(1.05)=\$52.50$. The opening reserve accumulated to year-end is $80(1.05)=\$84$. Substitution into the profit formula gives $$ g_0=52.50+40+100-50+84-0.02(3892.50)-0.10(40)-0.88(100). $$ The expected net claim is $0.02(3892.50)=\$77.85$; expected surrender-charge income is $0.10(40)=\$4$; and the expected closing reserve is $0.88(100)=\$88$. Therefore, $$ g_0=52.50+140-50+84-77.85-4-88=\$56.65. $$

Check: the account value is below the face benefit, so the assumed net claim is positive. Death, lapse, and persistence probabilities sum to $0.02+0.10+0.88=1$, and the profit calculation deducts the closing reserve only for policies that persist.

Common mistake avoided: applying the 5% interest rate after deducting the year-end charges. The charges are assessed after the account earns interest, so the account value is $(B_0+950)(1.05)-140$, not $(B_0+950-140)(1.05)$.

Comparing Two Persistency Assumptions

The following comparison uses the same contract and valuation assumptions for both scenarios. Mortality is $q_t=0.02$ in each year. Lapse rates are constant within each scenario: 0.10 for the lower-lapse scenario and 0.25 for the higher-lapse scenario. The second year is the final projection year, with $R_2=0$. Policy-year profits are paid at year-end and discounted at 5%.

Worked Example: Two-Year Profit Signature Under Alternative Lapse Rates

Use the contract data from the first example, with $R_0=\$80$, $R_1=\$100$, and $R_2=0$. Calculate annual profit per in-force policy, expected cohort profit by duration, and the two-year present value per policy issued under both lapse assumptions.

Method: keep the survivor’s account path fixed, but recalculate decrement probabilities and cohort counts. The year-1 account value is $A_0=\$6{,}107.50$. In year 2, the account starts at that value and follows the same account formula: $$ A_1=(6107.50+950)(1.05)-140 =7057.50(1.05)-140 =\$7{,}270.375. $$ Thus the year-2 net death claim is $10{,}000-7{,}270.375=\$2{,}729.625$.

In year 1, for lapse rate $w$, $p_0=0.98-w$. The per-policy profit is $$ g_0=52.50+140-50+84-0.02(3892.50)-40w-100(0.98-w) =\$50.65+\$60w. $$ For $w=0.10$, $p_0=0.88$ and $g_0=\$56.65$. For $w=0.25$, $p_0=0.73$ and $g_0=\$65.65$.

In year 2, the accumulated opening reserve is $100(1.05)=\$105$, and the closing reserve is zero. Therefore, $$ g_1=52.50+140-50+105-0.02(2729.625)-40w =\$192.9075-\$40w. $$ This gives $g_1=\$188.9075$ when $w=0.10$, and $g_1=\$182.9075$ when $w=0.25$.

Starting with one policy, the expected number in force at the start of year 2 is $n_1=p_0$. The year-2 cohort profits are therefore $0.88(188.9075)=\$166.2386$ and $0.73(182.9075)=\$133.522475$. The resulting comparison is:

Constant lapse rate	Year 1 cohort profit	Year 2 cohort profit	Two-year present value per policy issued
0.10	$56.65	$166.2386	$204.74
0.25	$65.65	$133.522475	$183.63

For the 0.10 lapse scenario, the present value is $56.65/1.05+166.2386/(1.05)^2=\$204.74$, rounded to the nearest cent. For the 0.25 lapse scenario, it is $65.65/1.05+133.522475/(1.05)^2=\$183.63$, also rounded to the nearest cent.

Check: higher lapse produces more year-1 profit per issued policy here, but it leaves fewer policies to generate year-2 profit. Across the full two-year projection, the lower-lapse scenario has the larger present value. This result depends on the stated charges, claims, and reserves; it is not a general rule that lower lapse always increases value.

Common mistake avoided: comparing only the per-policy profit in year 1. The cohort comparison must account for how many policies reach each duration. It is also incorrect to use $1-w$ as the persistence probability: mortality is a separate decrement, so $p=1-q-w$.

Interpreting an Intermediate Persistency Scenario

Worked Example: Projecting a Cohort at a 0.15 Lapse Rate

Using the same two-year contract and $q=0.02$, suppose the lapse rate is $w=0.15$. Starting with one policy, calculate the expected year-2 cohort profit and the present value of the two-year profit signature.

The year-1 persistence probability is $p_0=1-0.02-0.15=0.83$. From the year-1 expression above, profit per starting policy is $50.65+60(0.15)=\$59.65$. Year-2 profit per policy in force is $192.9075-40(0.15)=\$186.9075$. The expected year-2 cohort profit is $$ n_1g_1=0.83(186.9075)=\$155.133225. $$

Discount both end-year profits at 5%: $$ \operatorname{PV}=\frac{59.65}{1.05}+\frac{155.133225}{(1.05)^2} =\$56.8095+\$140.7104 =\$197.52. $$ The displayed components are rounded to four decimal places; the final present value is rounded to the nearest cent. The undiscounted total cohort profit is $59.65+155.133225=\$214.783225$.

Check: the expected year-2 count is less than one because $p_0=0.83$. Its profit present value, \$197.52, lies between the two scenario values in the previous comparison. This confirms the arithmetic for this intermediate lapse assumption, though it does not establish that profit must vary linearly with lapse.

Common mistake avoided: multiplying year-2 per-policy profit by the initial policy count of one. Only the expected 0.83 survivors contribute year-2 profit.

Exam Checks and Practical Interpretation

A reliable persistency comparison keeps several distinctions explicit. Account value is a policy-level balance; the risk reserve is a separate insurer liability. Profit per in-force policy is not the same as cohort profit, and neither is automatically the same as present value. The expected cohort size is determined by the probability of surviving all applicable decrements, not by lapse alone.

The examples also rely on a simplifying assumption that mortality and lapse rates do not depend on account value or policy characteristics. If a problem specifies selective lapses, changing mortality among persisting policies, a different death-benefit option, or a surrender value other than account value less a stated charge, the cash-flow model must be adjusted. In particular, the net death claim $F-A_t$ applies here only because the specified face benefit exceeds the account value.

State whether decrements are annual probabilities from the start of the year and verify $q_t+w_t+p_t=1$.
Apply interest and charges in the sequence specified by the contract.
Use the reserve at the correct duration and subtract the closing reserve only for policies that remain in force.
Weight each duration’s profit by its expected starting cohort size before calculating aggregate results.
Discount profits only after identifying their payment timing.

Key takeaway: Persistency affects both decrement cash flows and the number of contracts producing future profits. A sound universal life comparison projects the account, calculates profit per in-force policy, applies competing-decrement cohort weights, and then compares profits on a consistent timing and valuation basis.

Check Your Understanding

Use the stated timing conventions in each question. Amounts are per policy unless a cohort size is specified.

In a one-year model, the death probability is 0.012 and the lapse probability is 0.180. Find the probability of surviving and persisting to the next year. (Answer: 0.808)
A policy has starting account value $2,400, premium $600, premium load 0.04, effective annual interest 0.03, and year-end account charges of $30 and $45. Using the account recurrence in this tutorial, find the year-end account value. (Answer: $2,990.28)
For a year, accumulated premium-load income is $37.80, account charges total $80, expense is $40, accumulated opening reserve is $63, net death claim is $2,500, and closing reserve is $90. Let $q=0.01$, $w=0.14$, surrender charge $s=\$30$, and persistence probability $p=0.85$. Calculate expected profit per policy in force at the start of the year. (Answer: $35.10)
A cohort has 12,000 policies in force at the start of a year. If $q=0.02$ and $w=0.15$, how many policies are expected to persist to the next year, and what is the expected cohort profit if profit per in-force policy for the year is $75? (Answer: 9,960 policies; $747,000)
In the two-year example in this tutorial, use the constant lapse rate $w=0.25$. Calculate the present value per policy issued, using the expected cohort profits by duration and a 5% discount rate. (Answer: $183.63)

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