The Prospective Policy Value at a Duration

Why Value a Policy at a Duration?

A policy value at duration $t$ measures the insurer’s expected net obligation at that point in the contract. The policy has already been in force for $t$ years, so the relevant question is not what the insurer expected to pay or receive at issue. It is the present value, at duration $t$, of the benefits and premiums that remain, conditional on the policy being in force then.

The conditioning matters: a policy value is calculated for a policy that has survived and remains in force up to the valuation time. The calculation then accounts for the chance that the insured survives or dies after that time. This tutorial develops the definition and its application to discrete future cash flows, aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: The prospective policy value at duration $t$ is the conditional expected present value, at time $t$, of future benefits less the conditional expected present value of future premiums, given that the policy is in force at time $t$. Cash flows before or at the valuation time that have already occurred are not included as future cash flows.

Unless a problem states another convention, take the value just after any transaction scheduled at time $t$. Thus, a premium paid at time $t$ is not a future premium from this viewpoint. A premium due later is included only if it is still payable under the contract at that future date. A positive value indicates that the expected present value of future benefits exceeds that of future premiums.

Cash Flows, Conditioning, and Discounting

Let $i$ be the effective annual interest rate and $v=1/(1+i)$ the annual discount factor. At duration $t$, a payment made $j$ years later is discounted by $v^j$. For a discrete contract, let $B_j$ be the benefit amount paid at time $t+j$, with value zero if no benefit is paid then, and let $P_j$ be the premium amount due at that time, with value zero if no premium is due. Then the prospective value is

{}_tV = \sum_{j\geq 1} v^j E[B_j\mid \text{policy in force at }t] - \sum_{j\geq 1} v^j E[P_j\mid \text{policy in force at }t].

This expression is a bookkeeping framework: specify the possible payment dates, identify which events trigger each payment, and take the conditional expected value of each payment. It does not presume that every future benefit is paid or that every scheduled premium is received.

Set the valuation time.
List only cash flows that occur after duration $t$. Confirm whether a payment due exactly at $t$ has already been made.

Condition on being in force.
Use the mortality or other future-event probabilities conditional on the policy being in force at $t$, rather than probabilities from issue unless they are converted appropriately.

Weight each cash flow by its trigger.
A death benefit uses the probability of death in its covered period; a survival benefit or a premium conditional on survival uses the probability of surviving to its payment date.

Discount and take benefits minus premiums.
Discount each expected payment by the time from $t$ to its payment date, then subtract the premium present value from the benefit present value.

For example, suppose there is no lapse and the insured is age $x+t$ at duration $t$. Conditional on the policy being in force at $t$, the probability of surviving $j$ more years is ${}_jp_{x+t}$. The probability of dying during the $k$th future year is ${}_{k-1}p_{x+t}q_{x+t+k-1}$, where $q_y=1-p_y$. These probabilities are conditional on reaching duration $t$; they are not the unconditional probabilities from issue.

\begin{aligned} \text{Expected value at }t\text{ of a benefit }b_k \text{ paid at the end of future year }k &= v^k b_k\,{}_{k-1}p_{x+t}q_{x+t+k-1},\\ \text{Expected value at }t\text{ of a premium }Q_j \text{ due at }t+j\text{ if alive} &= v^j Q_j\,{}_jp_{x+t}. \end{aligned}

The second expression assumes the premium is due only if the insured survives to the payment date and the policy remains in force. If premiums are due under a different condition, that condition must replace the survival indicator in the expected value.

Why the Prospective Formula Holds

The formula follows from valuing each future contingent payment separately. For a payment $C_j$ made at $t+j$, its present value at $t$ is $v^jC_j$. If it is paid only on an event $A_j$, its amount can be written as $c_j I(A_j)$, where $I(A_j)$ is 1 when the event occurs and 0 otherwise. Conditional on the policy being in force at $t$,

E[v^j c_j I(A_j)\mid \text{in force at }t] = v^j c_j P(A_j\mid \text{in force at }t).

The present value of all future benefits is the sum of their individual present values, and the same is true for premiums. By linearity of conditional expectation, the expected present value of a sum is the sum of the conditional expected present values. Subtracting the premium sum from the benefit sum gives the stated policy value. This reasoning does not require the benefit events and premium events to be independent.

Worked Valuations at a Duration

Worked Example: Two-Year Term Coverage with a Future Premium

At duration $t$, a hypothetical term policy has two years of coverage remaining and pays $ \$10{,}000 $ at the end of the year of death. Conditional on being in force at $t$, the survival probabilities for the first and second future years are $0.90$ and $0.80$, respectively; the second probability is conditional on surviving the first year. The effective annual interest rate is $i=5\%$. A premium of $ \$500 $ is due at $t+1$ if the insured is alive then, and no other premiums remain. Find the prospective value.

Method. Split the benefit into death during the first future year and death during the second, then value the premium only if it is payable. The common mistake avoided here is applying an unconditional probability from issue rather than the probabilities conditional on being in force at $t$.

The probability of death in the first future year is $1-0.90=0.10$. The probability of death in the second future year is $0.90(1-0.80)=0.18$. With $v=1/1.05=0.95238095$, the expected present value of benefits is

\$10{,}000(0.10v+0.18v^2) = \$10{,}000\left(\frac{0.10}{1.05} +\frac{0.18}{1.05^2}\right) = \$2{,}585.03.

The premium is paid only if the insured survives to $t+1$. Its expected present value is

\$500(0.90)v = \$500\left(\frac{0.90}{1.05}\right) = \$428.57.

Therefore, the prospective value is $ \$2{,}585.03-\$428.57=\boxed{\$2{,}156.46} $. As a check, the benefit calculation includes exactly the two possible years of death, and the premium is weighted by the $0.90$ probability of surviving to its due date. Amounts are rounded to the nearest cent only at the final step.

Worked Example: Two-Year Endowment with a Premium Due After One Year

A hypothetical endowment policy has two years remaining. It pays $ \$6{,}000 $ at the end of the year of death, if death occurs during those two years, or $ \$6{,}000 $ at the end of year two if the insured survives to maturity. Conditional survival probabilities for years one and two are $0.90$ and $0.80$, respectively. The effective annual interest rate is $i=0\%$. A premium of $ \$700 $ is due at $t+1$ if the insured is alive then; the premium at $t$ has already been paid. Find the value at $t$.

Method. Partition the future into death in year one, death in year two, and survival to maturity. These are mutually exclusive and exhaustive outcomes. The trap is treating the premium as certain even though it is due only if the insured survives to $t+1$.

The three outcome probabilities are

\begin{aligned} P(\text{death in year 1}\mid\text{in force at }t)&=1-0.90=0.10,\\ P(\text{death in year 2}\mid\text{in force at }t)&=0.90(1-0.80)=0.18,\\ P(\text{survival to maturity}\mid\text{in force at }t)&=0.90(0.80)=0.72. \end{aligned}

They sum to $0.10+0.18+0.72=1.00$, so exactly one of the three benefit outcomes occurs. Each outcome pays $ \$6{,}000 $, and no discounting is required at $i=0\%$. Thus the expected present value of benefits is $ \$6{,}000(1.00)=\$6{,}000 $.

The premium is payable with probability $0.90$, and its due date is one year after $t$. Since $v=1$, its expected present value is $ \$700(0.90)(1)=\$630 $. Hence the prospective value is

\$6{,}000-\$630=\boxed{\$5{,}370.00}.

The answer is checked by separating the guaranteed benefit amount across all possible outcomes from the contingent premium: the benefit is paid in every outcome, whereas the premium is paid only in the first-year survival outcomes.

Worked Example: Death and Maturity Benefits with Two Remaining Premiums

At duration $t$, a hypothetical two-year policy pays $ \$4{,}000 $ at the end of the year of death if death occurs during the next two years, or $ \$2{,}000 $ at $t+2$ if the insured survives to that date. The conditional survival probabilities for the next two years are $0.90$ and $0.80$, with the second conditional on surviving year one. The effective annual interest rate is $i=0\%$. Premiums of $ \$300 $ at $t+1$ and $ \$200 $ at $t+2$ are each due only if the insured is alive at their respective due dates. Find the value just before any payment at $t+2$.

Method. Value each benefit and premium by its own payment event. The common error avoided is using the year-one survival probability for the premium due at $t+2$, which requires survival through both years.

The probability of death in year one is $0.10$, and the probability of death in year two is $0.90(0.20)=0.18$. The probability of survival to $t+2$ is $0.90(0.80)=0.72$. The expected benefit value is therefore

\$4{,}000(0.10)+\$4{,}000(0.18)+\$2{,}000(0.72) = \$400+\$720+\$1{,}440 = \$2{,}560.

The expected value of the premium at $t+1$ is $ \$300(0.90)=\$270 $. The expected value of the premium at $t+2$ is $ \$200(0.72)=\$144 $. At zero interest the discount factors are both 1, so the total expected value of premiums is $ \$270+\$144=\$414 $. The prospective value is

\$2{,}560-\$414=\boxed{\$2{,}146.00}.

The probabilities for the two death outcomes and survival to maturity sum to $0.10+0.18+0.72=1.00$. The premiums use the survival probabilities to their own due dates, confirming that the timing and contingencies have been applied consistently.

Common Errors and a Final Check

The policy value is prospective, so the calculation starts at the duration being valued and looks forward. A cash flow already paid is not subtracted again, and a premium due in the future is not automatically certain. For each future payment, ask both when is it paid? and what event makes it payable? Those answers determine its discount factor and its probability weight.

Using probabilities from issue without conditioning. At duration $t$, use future probabilities conditional on the policy being in force at $t$.
Including a premium already paid. A premium paid by the valuation time is not a future premium.
Assuming every future premium is received. If payment requires survival to its due date, weight it by the probability of survival to that date.
Discounting from issue instead of from duration $t$. A payment $j$ years after the valuation time is discounted by $v^j$, not by a factor based on the full policy term from issue.
Subtracting premiums from benefits before applying their contingencies. First calculate each expected present value using its own event and timing; then subtract the premium total from the benefit total.

Key takeaway: At a duration, a prospective policy value is the conditional expected present value of the remaining benefits minus the conditional expected present value of the remaining premiums. Correct conditioning, payment timing, and discounting are all essential parts of the definition.

Check Your Understanding

For each problem, value the future cash flows from the stated duration and condition on the policy being in force then. Round monetary answers to the nearest cent.

A two-year term policy pays $ \$10{,}000 $ at the end of the year of death. Conditional survival probabilities for years one and two are $0.80$ and $0.75$. The interest rate is $0\%$, and a $ \$400 $ premium is due at $t+1$ if alive. Find the value. (Answer: $ \$3{,}680.00 $)
A two-year pure endowment pays $ \$5{,}000 $ at $t+2$ only if the insured survives to then. Conditional survival probabilities are $0.90$ for year one and $0.80$ for year two. The effective annual interest rate is $10\%$. A $ \$300 $ premium is due at $t+1$ if alive. Find the value. (Answer: $ \$2{,}729.75 $)
A two-year term policy pays $ \$8{,}000 $ at the end of the year of death. Conditional survival probabilities are $0.95$ and $0.90$. The effective annual interest rate is $4\%$, and a $ \$500 $ premium is due at $t+1$ if alive. No other premiums remain. Find the value. (Answer: $ \$630.55 $)
A two-year policy pays $ \$4{,}000 $ at the end of the year of death if death occurs during the term, or $ \$5{,}000 $ at $t+2$ if the insured survives. Conditional survival probabilities are $0.80$ and $0.75$, and the interest rate is $0\%$. Premiums of $ \$300 $ at $t+1$ and $ \$200 $ at $t+2$ are each due only if alive on their due dates. Find the value. (Answer: $ \$4{,}240.00 $)

What You'll Learn