Persistency-Weighted Expected Present Values

Why Persistency Weighting Matters

A universal life policy’s future cash flows depend on whether it remains in force. A premium due at the start of a year is collected only if the policy is active then. A death benefit is paid only if death occurs while the policy is active, and a surrender payment is made only if lapse occurs. Consequently, valuing a cash flow as though every policy persists to every future duration generally gives the wrong issue-date expected present value.

The key distinction is between a rate conditional on being active at the start of a year and the probability, measured at issue, that the corresponding cash flow occurs. Persistency weighting converts the first quantity into the second. Discounting then reflects when the payment is made.

Definition: Consider a discrete-time model with annual periods. At the start of year $t+1$, a policy is active unless it has already terminated. Conditional on being active at that start, let $q_t$ be the probability of death during the year, $l_t$ the probability of lapse during the year, and $s_t=1-q_t-l_t$ the probability of remaining active through the year. Assume death and lapse are mutually exclusive exits within each period. Let $p_t$ be the issue-date probability that the policy is active at time $t$, with $p_0=1$. Then $p_{t+1}=p_ts_t$. The issue-date probabilities of death and lapse during year $t+1$ are $p_tq_t$ and $p_tl_t$, respectively.

This convention treats an active-at-time-$t$ cash flow as payable at the start of period $t+1$, and death or lapse payments during that period as payable at its end. A different contract or valuation basis may specify different timing; in that case, use the stated timing rather than forcing it into this convention.

With an effective annual interest rate $i$, the discount factor for a payment at time $t$ is $v^t$, where $v=1/(1+i)$. A cash flow of amount $C_t$, payable at time $t$ only if a specified event occurs, contributes $v^t C_t$ multiplied by the issue-date probability of that event. For an outflow, the contribution is a cost; for an inflow, it is a receipt.

From Conditional Rates to Issue-Date Probabilities

The probability $s_t$ is conditional on the policy having reached the start of year $t+1$. To reach the start of a later year, the policy must persist through every preceding year. Repeated conditional multiplication therefore gives:

p_t=\prod_{j=0}^{t-1}s_j =\prod_{j=0}^{t-1}(1-q_j-l_j), \qquad p_0=1.

For example, if the conditional probability of remaining active is $0.90$ in each of two successive years, the probability of being active at the start of the third year is $0.90\times0.90=0.81$, not $0.90$. Each year’s death and lapse rates are conditional rates, not issue-date probabilities.

Theorem: If an amount $C_t$ is payable at time $t$ whenever the policy is active at that time, its issue-date expected present value is $v^t p_t C_t$. If an amount $B_{t+1}$ is payable at the end of year $t+1$ on death during that year, its issue-date expected present value is $v^{t+1}p_tq_tB_{t+1}$. Replace $q_t$ by $l_t$ for a lapse payment, provided the payment is made on lapse during that year.

These formulas require the stated payment trigger and timing. For instance, a charge deducted at the start of the year is weighted by $p_t$; a charge made only at the end of the year if the policy remains active is weighted by $p_ts_t=p_{t+1}$. Applying one weight to both cash flows simply because they are associated with the same year ignores the difference in timing and eligibility.

Why the Weighting Formula Holds

An issue-date present value is a random variable: the cash flow may or may not occur, depending on the policy’s future path. Let $I_t$ equal 1 if the policy is active at time $t$, and 0 otherwise. A time-$t$ payment $C_t$, due only when active, has present value $v^tC_tI_t$. Taking expectations gives:

$$ E[v^tC_tI_t] =v^tC_tE[I_t] =v^tC_tP(I_t=1) =v^tC_tp_t. $$

For a death payment in year $t+1$, define an indicator that equals 1 if the policy is active at the start of the year and death occurs during it. Its probability is $p_tq_t$: the policy must first be active at the start, then death must occur conditional on that status. Multiplying the benefit by this probability and discounting to the end of the year gives $v^{t+1}p_tq_tB_{t+1}$. This argument is just the expectation of a payment amount multiplied by an event indicator; it does not require the cash flow amounts in different years to be independent.

If the policy has several possible cash flows, expected present values can be calculated one component at a time and added. This follows from the linearity of expectation. Premiums or charges are not automatically netted against benefits before weighting: their eligibility events and payment times may differ.

Worked Example: Weighting Premiums, Expenses, and Exit Payments

A hypothetical two-year policy has an effective annual interest rate of $i=5\%$. Conditional on being active at the start of year 1, the death and lapse probabilities are $q_0=0.02$ and $l_0=0.08$. For year 2 they are $q_1=0.03$ and $l_1=0.07$. A premium of $2,000 and an expense of $300 are payable at the start of each year if the policy is active. A $100,000 death benefit or a $5,000 surrender payment is payable at the end of the year of death or lapse. Calculate the issue-date expected present value of premiums less expenses and exit payments. Use the method of weighting each cash flow by its issue-date event probability.

Step 1: Find persistence probabilities. The conditional persistence probabilities are $s_0=1-0.02-0.08=0.90$ and $s_1=1-0.03-0.07=0.90$. Thus $p_0=1$ and $p_1=1(0.90)=0.90$.

Step 2: Find issue-date probabilities of exit. In year 1, death and lapse probabilities measured from issue are $1(0.02)=0.02$ and $1(0.08)=0.08$. In year 2, they are $0.90(0.03)=0.027$ and $0.90(0.07)=0.063$.

Step 3: Calculate premium and expense present values. Year 1 cash flows are at time 0. Year 2 cash flows are at time 1 and are due only if active then. The effective annual discount factors are $v=1/1.05$ and $v^2=1/1.1025$. Therefore:

\begin{aligned} \text{EPV of premiums} &=2000+\frac{0.90(2000)}{1.05} =2000+1714.2857 =\$3,714.29,\\ \text{EPV of expenses} &=300+\frac{0.90(300)}{1.05} =300+257.1429 =\$557.14. \end{aligned}

Step 4: Calculate death and lapse payments. These payments occur at the end of their respective years, so year 1 payments are discounted one year and year 2 payments two years:

\begin{aligned} \text{EPV of death benefits} &=\frac{0.02(100000)}{1.05} +\frac{0.027(100000)}{1.1025}\\ &=1904.7619+2448.9796 =\$4,353.74,\\ \text{EPV of surrender payments} &=\frac{0.08(5000)}{1.05} +\frac{0.063(5000)}{1.1025}\\ &=380.9524+285.7143 =\$666.67. \end{aligned}

The requested net expected present value is premium receipts less the expense and exit-payment costs: $3714.2857-557.1429-4353.7415-666.6667=-1863.2654$. Rounded to the nearest cent, this is −$1,863.27. The negative value means the specified expected outflows exceed the specified expected premium receipts on this basis. The calculation avoids the mistake of applying the year 2 conditional rates directly as issue-date probabilities.

Conditional Valuation and the Issue-Date Basis

Sometimes a value is first calculated at a future duration, conditional on the policy being active then. To convert that value to an issue-date expected present value, multiply by the probability of reaching that duration and discount from that duration to issue. The conditional value itself does not include the probability of having persisted to that point.

Worked Example: Converting a Conditional Value to Issue

A hypothetical policy is active at issue. In year 1, its conditional probabilities of death and lapse are $0.10$ and $0.10$. Thus the probability it is active at time 1 is $0.80$. Conditional on being active at time 1, in year 2 the probabilities of death and lapse are $0.20$ and $0.10$. At time 1, a $500 charge is due if active. At the end of year 2, a $40,000 death payment or a $3,000 lapse payment is due on the corresponding exit. With $i=4\%$, find the issue-date expected present value of these costs.

Conditional calculation at time 1. Given the policy is active at time 1, the time-1 charge is certain. The year 2 death and lapse payments have conditional probabilities $0.20$ and $0.10$, and are paid one year later. Thus the expected present value at time 1, conditional on being active then, is:

500+\frac{0.20(40000)+0.10(3000)}{1.04} =500+\frac{8300}{1.04} =500+7980.7692 =\$8,480.77.

This is not yet an issue-date value. The probability of reaching time 1 is $0.80$, and time 1 is one year after issue. Weight and discount the conditional value:

\frac{0.80(8480.7692)}{1.04} =\$6,523.67.

Check by direct weighting. The charge has issue-date present value $0.80(500)/1.04=\$384.62$. The issue-date probabilities of year 2 death and lapse are $0.80(0.20)=0.16$ and $0.80(0.10)=0.08$. The two end-of-year-2 payments therefore have present value $(0.16(40000)+0.08(3000))/1.04^2=\$6,139.05$. Their sum is $6,523.67, agreeing with the conditional method. The common mistake is to use a time-1 conditional value as if it were already measured from issue, omitting both the persistence weight and the first year of discounting.

Many Cash Flows: A Consistent Projection

For a multi-year policy, the safest procedure is to make a separate row for each type of cash flow and specify its eligibility event and payment time. In particular, record whether the cash flow is payable at the start of a period, at its end if still active, or at the end on a specified exit. Then calculate persistence and event probabilities before applying discount factors.

Set the period convention.
State the valuation date, effective annual interest rate, conditional death and lapse probabilities, and when each cash flow is due.

Project in-force probabilities.
Begin with $p_0=1$, then calculate $p_{t+1}=p_t(1-q_t-l_t)$ for each year needed.

Weight by the trigger event.
Use $p_t$ for a start-of-year payment if active, $p_tq_t$ for a death payment, and $p_tl_t$ for a lapse payment. For an end-of-year payment conditional on persistence, use $p_{t+1}$.

Discount and combine.
Discount each weighted amount from its actual payment time to issue. Add receipts and costs separately before forming any requested net amount.

Worked Example: A Three-Year Issue-Date Projection

A hypothetical policy has an effective annual interest rate of $i=4\%$. Its conditional death and lapse probabilities are $(q_0,l_0)=(0.01,0.04)$, $(q_1,l_1)=(0.02,0.05)$, and $(q_2,l_2)=(0.03,0.07)$. A $1,500 premium and $400 expense are payable at the start of each year if active. Death pays $60,000 and lapse pays $4,000 at the end of the year of exit. Find the issue-date expected present value of premiums less expenses and exit payments. Use the period-by-period projection method.

Step 1: Calculate persistence. The persistence probabilities by year are $s_0=0.95$, $s_1=0.93$, and $s_2=0.90$. Hence $p_0=1$, $p_1=0.95$, and $p_2=0.95(0.93)=0.8835$. The probabilities of death in years 1, 2, and 3 are $0.01$, $0.95(0.02)=0.019$, and $0.8835(0.03)=0.026505$. The corresponding lapse probabilities are $0.04$, $0.95(0.05)=0.0475$, and $0.8835(0.07)=0.061845$.

Step 2: Value start-of-year premiums and expenses. The start-of-year probabilities are $1$, $0.95$, and $0.8835$, at times 0, 1, and 2. The discount factors at those times are $1$, $1/1.04$, and $1/1.0816$. Thus:

\begin{aligned} \text{EPV of premiums} &=1500\left(1+\frac{0.95}{1.04}+\frac{0.8835}{1.0816}\right)\\ &=1500(1+0.9134615+0.8168454) =\$4,095.46,\\ \text{EPV of expenses} &=400(1+0.9134615+0.8168454) =\$1,092.12. \end{aligned}

Step 3: Value exit payments. Exit payments occur at times 1, 2, and 3, with discount factors $1/1.04$, $1/1.0816$, and $1/1.124864$:

\begin{aligned} \text{EPV of death payments} &=60000\left( \frac{0.01}{1.04} +\frac{0.019}{1.0816} +\frac{0.026505}{1.124864}\right)\\ &=60000(0.0096154+0.0175666+0.0235628) =\$3,044.69,\\ \text{EPV of lapse payments} &=4000\left( \frac{0.04}{1.04} +\frac{0.0475}{1.0816} +\frac{0.061845}{1.124864}\right)\\ &=4000(0.0384615+0.0439164+0.0549800) =\$549.43. \end{aligned}

Step 4: Form and check the net amount. Premiums less all specified costs are $4095.46-1092.12-3044.69-549.43=-590.78$. The issue-date net expected present value is therefore −$590.78, rounded to the nearest cent. As a check, the EPV of premiums less expenses is $3,003.34, while the combined death and lapse payment EPV is $3,594.12; their difference is −$590.78. The calculation avoids treating conditional year 2 or year 3 exit rates as though every policy had reached those years.

Common Traps and Interpretation

Persistency weighting is a valuation technique, not a claim that a particular policy follows an average path. It averages cash flows across possible paths using their probabilities. The resulting expected present value depends on the assumptions, cash-flow definitions, and timing specified in the model.

Using conditional rates as issue-date probabilities. A year 3 death probability is $p_2q_2$, not just $q_2$.
Using the wrong in-force probability for a cash flow. A start-of-year premium uses $p_t$; a year-end charge payable only if still active uses $p_{t+1}$.
Discounting to the wrong time. A payment at the end of year $t+1$ is discounted by $v^{t+1}$, not $v^t$.
Adding mutually exclusive exit probabilities incorrectly. Under the stated model, $q_t+l_t+s_t=1$ for each year. The in-force probability at the next year is $p_ts_t$.
Mixing conditional and unconditional values. A value conditional on being active at time $t$ must be weighted by $p_t$ and discounted from time $t$ when converted to an issue-date value.

This framework is useful whenever a long-term product has cash flows contingent on continued coverage or termination. It makes the valuation basis transparent: assumptions determine the probability of each event, contract terms determine the cash flow and its timing, and the interest rate determines its discounting. The approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated discrete-time convention and round dollar answers to the nearest cent.

A policy has conditional persistence probabilities $0.92$, $0.90$, and $0.95$ in its first three years. What is the issue-date probability that it is active at the start of year 4? (Answer: 0.7866)
A policy has $i=5\%$, is active at the start of year 2 with probability $0.90$, and has a conditional year 2 death probability of $0.04$. A $25,000 benefit is paid at the end of year 2 on death. Find its issue-date expected present value. (Answer: $816.33)
With $i=5\%$, a premium of $800 is due at times 0, 1, and 2 if the policy is active. Its probabilities of being active at those times are $1$, $0.90$, and $0.72$. Find the issue-date expected present value of the premiums. (Answer: $2,008.16)
A policy has $i=4\%$. Its lapse probabilities conditional on being active at the start of years 1 and 2 are $0.06$ and $0.08$, and it persists through year 1 with probability $0.94$. A $2,000 surrender payment is made at each year-end lapse. Find the issue-date expected present value of the two possible surrender payments. (Answer: $254.44)
A policy has $i=5\%$, with $(q_0,l_0)=(0.02,0.08)$, $(q_1,l_1)=(0.03,0.07)$, and $(q_2,l_2)=(0.04,0.05)$. At the start of each of three years, if active, it receives a $500 premium and incurs a $100 expense. At the end of each year, death pays $10,000 and lapse pays $1,000. Find the issue-date expected present value of premiums less expenses and exit payments. (Answer: $153.16)

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