Persistency Assumptions and Profit Signatures

Why Persistency Changes a Profit Signature

A duration profit does not contribute to the expected value of a policy issued if the policy is no longer in force when that profit is earned. Persistency assumptions determine the probability of reaching each duration, so changing them changes the weights applied to the duration profits. The effect on NPV is not automatically positive: retaining more policies can preserve later gains, but it can also preserve later losses.

The calculations below isolate lapse or withdrawal persistency. They use a hypothetical portfolio and a simplified model in which each duration profit is earned only if the policy remains in force to the end of that duration. In a full profit test, the profit at a duration may also depend on other assumptions and cash flows; the timing and conditions for each amount must be stated before selecting its weight.

Definition: Let $p_t$ be the probability that a policy in force at the start of duration $t$ remains in force to its end. The corresponding lapse probability during that duration is $1-p_t$. Let $g_t$ be the expected profit amount at the end of duration $t$, conditional on the policy remaining in force to that time. With $w_0=1$, the probability of remaining in force through duration $t$ is $w_t=\prod_{j=1}^{t}p_j$. The persistency-weighted profit signature amount is $s_t=w_tg_t$.

For effective annual interest rate $i$, let $v=1/(1+i)$. Under this timing convention, the discounted contribution at duration $t$ is $v^ts_t$, and the expected NPV per policy issued over $n$ durations is

\operatorname{NPV}=\sum_{t=1}^{n}v^tw_tg_t =\sum_{t=1}^{n}v^t\left(\prod_{j=1}^{t}p_j\right)g_t.

The profit signature amounts $s_t$ are not yet discounted; the discounted contributions are $v^ts_t$. This distinction makes it easier to see whether a change comes from persistency weights or from interest discounting.

Why the Persistency Weights Multiply

To reach the end of duration 2, a policy must remain in force through duration 1 and then through duration 2. More generally, being in force through duration $t$ requires passing each of the successive in-force conditions. By the multiplication rule for conditional probabilities,

P(\text{in force through }t) =\prod_{j=1}^{t}P(\text{in force through }j\mid \text{in force through }j-1) =\prod_{j=1}^{t}p_j.

This argument does not require the year-by-year persistency events to be unconditionally independent: each $p_j$ is explicitly conditional on being in force at the start of duration $j$. If $g_t$ is the conditional expected profit at the end of duration $t$, multiplying it by $w_t$ gives the expected duration-$t$ profit per policy issued. Discounting that expected amount by $v^t$, then adding over durations, gives the NPV formula.

The model has an important boundary. It does not include a payment made specifically because a policy lapses. If a lapse during duration $t$ generates an additional profit amount $l_t$, and that amount is paid at the end of the duration, its probability weight is $w_{t-1}(1-p_t)$. Its discounted expected contribution is $v^tw_{t-1}(1-p_t)l_t$. In that case, include this lapse-event contribution as well as any in-force profit; do not treat it as though it were earned only by policies persisting to the end.

Worked Example: Finding the Break-Even Persistency Level

A hypothetical two-year contract has no discounting. It produces a profit of $-\$60$ at the end of year 1 if the policy remains in force through that year, and a profit of $\$100$ at the end of year 2 if it remains in force through both years. Persistency is the same in each year, with annual probability $p$. Find the NPV as a function of $p$, its break-even values, and the NPV at $p=0.80$.

Method. First calculate the probability weight for each duration. The year-1 weight is $w_1=p$. The year-2 weight is $w_2=p\times p=p^2$. With no discounting, the NPV is therefore

\operatorname{NPV}(p)=(-60)p+(100)p^2 =p(100p-60).

Break-even occurs when $p(100p-60)=0$. The two solutions are $p=0$ and $p=0.60$. They correspond to annual lapse probabilities of $1-0=1.00$, or 100%, and $1-0.60=0.40$, or 40%, respectively. The first solution is valid: if no policy persists through the first year, neither of the specified end-of-year profits is earned.

At $p=0.80$, the year-1 weighted profit is $-\$60(0.80)=-\$48$, and the year-2 weighted profit is $\$100(0.80^2)=\$64$. Thus $\operatorname{NPV}=-\$48+\$64=\$16$. The calculation checks: $0.80(100(0.80)-60)=0.80(20)=\$16$.

Common mistake avoided. Treating the year-2 weight as $p$ would give $-60p+100p$, incorrectly applying only one year of persistency to a profit that requires remaining in force through two years.

Recalculating a Signature Under Alternative Assumptions

When comparing persistency bases, hold the duration profit vector and interest assumption fixed unless the problem explicitly changes them. Recalculate the in-force weights under each basis, multiply each weight by the corresponding duration profit, and then discount. This separates the effect of persistency from changes in product design or interest.

Worked Example: Higher Persistency with Later Positive Profits

A hypothetical two-year product has conditional end-of-year profits $g_1=-\$110$ and $g_2=\$121$. The effective annual interest rate is 10%, so $v=1/1.10$. Compare NPV under persistency assumptions $(p_1,p_2)=(0.80,0.75)$ and $(0.90,0.90)$.

Method and first basis. Under $(0.80,0.75)$, the weights are $w_1=0.80$ and $w_2=0.80(0.75)=0.60$. The persistency-weighted signature amounts are

s_1=(-110)(0.80)=-88,\qquad s_2=(121)(0.60)=72.60.

The discounted contributions are $-88/1.10=-\$80.00$ and $72.60/1.10^2=72.60/1.21=\$60.00$. Therefore the NPV is $-\$80+\$60=-\$20.00$.

Second basis. Under $(0.90,0.90)$, $w_1=0.90$ and $w_2=0.90(0.90)=0.81$. The signature amounts are $(-110)(0.90)=-\$99.00$ and $(121)(0.81)=\$98.01$. Discounting gives $-99/1.10=-\$90.00$ and $98.01/1.21=\$81.00$. The NPV is $-\$90+\$81=-\$9.00$.

The higher-persistency basis increases NPV by $-\$9-(-\$20)=\$11.00$. It increases the expected early loss as well as the later gain, but the added weight on the positive second-year profit is more valuable in this example. The two NPVs are checked directly against the discounted signature contributions.

Common mistake avoided. The second-year weight under the first basis is not $p_2=0.75$ alone. Using that incorrect weight gives a second-year signature amount of $121(0.75)=\$90.75$ and a discounted contribution of $\$75.00$, leading to the wrong NPV of $-\$5.00$. The correct weight is $p_1p_2=0.60$, because the policy must persist through both years.

Persistency Can Also Reduce NPV

The direction of the NPV change depends on the duration-profit pattern, not on whether persistency is described as “better.” If early profits are positive and a later profit is negative, greater persistency can retain more policies long enough to generate that later loss. The following example uses the same two persistency bases and interest rate to isolate the change in profit timing and sign.

Worked Example: Higher Persistency with a Later Loss

A hypothetical contract has conditional end-of-year profits $g_1=\$100$ and $g_2=-\$121$. The effective annual interest rate is 10%. Compare its NPV under persistency assumptions $(0.80,0.75)$ and $(0.90,0.90)$.

Under the first basis, $w_1=0.80$ and $w_2=0.80(0.75)=0.60$. The signature amounts are $100(0.80)=\$80.00$ and $(-121)(0.60)=-\$72.60$. Their discounted contributions are $80/1.10=\$72.7273$ and $-72.60/1.21=-\$60.00$. Before final rounding, the NPV is $72.727272\ldots-60=\$12.727272\ldots$, or \$12.73 to the nearest cent.

Under the second basis, $w_1=0.90$ and $w_2=0.90(0.90)=0.81$. The signature amounts are $\$90.00$ and $-\$98.01$. The discounted contributions are $90/1.10=\$81.818181\ldots$ and $-98.01/1.21=-\$81.00$. The NPV is $\$0.818181\ldots$, or \$0.82 to the nearest cent.

Here, moving to higher persistency reduces NPV by approximately $\$11.91$ per policy issued: $0.818181\ldots-12.727272\ldots=-11.909090\ldots$. Greater persistency gives more policies access to the early gain, but it also causes more policies to remain for the larger later loss. The calculation checks because the direction of the change is consistent with the weighted, discounted contributions at each duration.

Interpreting Persistency Comparisons

A reliable comparison starts with the conditions under which each profit is earned. In the model used here, every $g_t$ is an end-of-duration amount conditional on being in force at that time. If a problem instead states that a profit is earned at the beginning of a duration, or that a lapse itself produces a cash flow, the weights and timing must reflect that description. Applying end-of-year persistency weights to a beginning-of-year amount, or omitting a lapse-triggered amount, changes the model rather than merely simplifying its arithmetic.

For two persistency bases $A$ and $B$ applied to the same product and interest rate, the NPV difference is

\operatorname{NPV}_B-\operatorname{NPV}_A =\sum_{t=1}^{n}v^t(w_{t,B}-w_{t,A})g_t.

This identity follows by subtracting the two NPV sums term by term; the interest factor and duration profit are unchanged, so only the weights differ. If basis $B$ has greater weights at all durations, then each difference $w_{t,B}-w_{t,A}$ is nonnegative. That does not determine the sign of the total difference when some $g_t$ are negative and others positive. Positive duration profits contribute positively to the difference, while negative duration profits contribute negatively.

This is why a persistency sensitivity is not a general statement that “lower lapses improve profitability.” Its result depends on the product’s profit emergence, the timing of the cash flows, and the way the profit vector is defined. A useful check is to inspect the sign of each duration profit alongside the change in its weight before interpreting the final NPV.

Key takeaway: For an end-of-duration profit conditional on remaining in force, multiply by the cumulative persistency probability through that duration, then discount. Higher persistency raises NPV when its added weight falls mainly on positive contributions, and can lower NPV when it adds sufficient weight to negative contributions.

Check Your Understanding

For each question, use the stated timing convention: duration profits are earned at year-end only if the policy remains in force to that year-end. Give money answers to the nearest cent where needed.

A one-year contract has a year-end profit of $-\$40$ conditional on remaining in force, persistency $p_1=0.85$, and no discounting. Find its signature amount and NPV. (Answer: signature $-\$34.00$; NPV $-\$34.00$)
A two-year contract has profits $g_1=\$50$, $g_2=-\$80$, persistencies $p_1=0.80$, $p_2=0.90$, and no discounting. Find both signature amounts and the NPV. (Answer: $s_1=\$40.00$, $s_2=-\$57.60$; NPV $-\$17.60$)
With no discounting, a two-year contract has profits $-\$30$ and $\$50$, and the same annual persistency $p$ in both years. Find every persistency probability in $[0,1]$ that gives zero NPV. (Answer: $p=0$ and $p=0.60$, corresponding to annual lapse probabilities 100% and 40%)
A one-year contract pays a conditional year-end profit of $\$100$. Persistency is $0.90$, and the effective annual interest rate is 5%. Find its discounted NPV per policy issued. (Answer: $\$85.71$)
A three-year contract has profits $(\$20,-\$10,\$40)$, persistencies $(0.90,0.80,0.50)$, and no discounting. Find the three signature amounts and NPV. (Answer: $s_1=\$18.00$, $s_2=-\$7.20$, $s_3=\$14.40$; NPV $\$25.20$)

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