Single-Premium Policy Values

Why Single-Premium Policy Values Are Direct

A single-premium contract has a simpler prospective value than a contract with continuing premiums: after issue, there are no future premium payments to subtract. The value at a duration is the conditional expected present value of the benefits still payable, given that the insured is alive at that duration. This direct link between the reserve and future benefits is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

The timing convention matters. In this tutorial, benefits are paid at the end of the year of death. For an endowment, the same amount is paid at the end of the term if the insured survives. Values at duration $k$ are taken immediately after the $k$-th policy anniversary, conditional on survival to that time, and before any benefit due at a later time.

Definition: For a single-premium contract with no future premiums, the prospective policy value at duration $k$, conditional on survival to that duration, is the expected present value at time $k$ of benefits payable after time $k$. In a net valuation with no future expenses, it is the conditional expected present value of those future benefits. At issue, the value is the net single premium.

Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $q_{x+t}$ be the probability that a life aged $x+t$ dies within the following year, and let $p_{x+t}=1-q_{x+t}$. Thus ${}_jp_x$ is the probability of surviving $j$ years from age $x$. We use a fixed benefit amount $B$, annual discrete benefits, and a mortality model specified by the relevant $q$-values. Unless stated otherwise, premiums and expenses other than the single premium are excluded.

Prospective Values for Discrete Benefits

At duration $k$, the probability of dying in the $j$-th year after the valuation time is ${}_{j-1}p_{x+k}q_{x+k+j-1}$. A benefit paid at the end of that year is discounted $j$ years to duration $k$. Adding the discounted expected payments gives the prospective value.

{}_kV_x^{\text{WL}} = B\sum_{j=1}^{\infty}v^j\,{}_{j-1}p_{x+k}q_{x+k+j-1}.

For an $n$-year term policy, only deaths during the remaining $n-k$ years are covered, so the sum stops at $n-k$. For an $n$-year endowment with benefit $B$ payable on death during the term or on survival to its end, include the survival benefit as well:

{}_kV_x^{\text{End}} = B\left[ \sum_{j=1}^{n-k}v^j\,{}_{j-1}p_{x+k}q_{x+k+j-1} + v^{n-k}\,{}_{n-k}p_{x+k} \right], \qquad 0\le k<n.

The first part of the endowment formula covers death benefits. The final term is the discounted survival benefit. For a term policy, that final term is absent: survival to the end of the term does not produce a payment.

Identify the valuation time.
At duration $k$, work from age $x+k$, conditional on survival to that point.

List future payment events.
Include each covered year of death and, for an endowment, survival to the maturity date.

Weight and discount each payment.
Use its conditional probability from duration $k$ and discount by the number of years from $k$ to payment.

Sum the values and check the contract.
A term policy has no maturity payment; an endowment does. At issue, the value is the net single premium.

Why the Formula Works

The prospective value is an expectation over mutually exclusive future payment events. For a whole-life policy valued at duration $k$, the event “death in year $j$ after duration $k$” has conditional probability ${}_{j-1}p_{x+k}q_{x+k+j-1}$. On that event, the amount $B$ is paid $j$ years later and has value $Bv^j$ at duration $k$. Multiplying payment by event probability and adding across the possible years gives the whole-life sum. For an endowment, survival to the end of the remaining term is an additional, disjoint event; its probability is ${}_{n-k}p_{x+k}$ and its payment value is $Bv^{n-k}$. This also explains why probabilities must be conditional on being alive at the valuation time.

A useful special case is a whole-life policy with a constant annual death probability $q$, so $p=1-q$ each year. The probability of death in year $j$ is $p^{j-1}q$, and the present-value terms form a geometric series:

A_x = \sum_{j=1}^{\infty}v^jp^{j-1}q = vq\sum_{j=0}^{\infty}(vp)^j = \frac{vq}{1-vp}.

The series converges when $vp<1$, which holds when $i>0$ and $0\le p\le1$. The formula depends on constant mortality across years; it is not appropriate when the annual death probabilities vary by age.

Worked Example: Whole-Life Value with Changing Mortality

A hypothetical single-premium whole-life policy pays $10,000 at the end of the year of death. Assume $i=5\%$. The annual death probabilities are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+t}=0.25$ for every $t\ge2$. Find the value at issue and the value at duration 2, conditional on survival to duration 2.

Method. Use the conditional death-benefit sum. For duration 2, mortality is a constant $q=0.25$ from then on, so the geometric-series result can be used for the remaining whole-life value. This avoids applying one constant-$q$ formula to the entire policy when the first two years have different mortality.

Here $v=1/1.05=20/21$. At duration 2, $p=0.75=3/4$, so:

A_{x+2} = \frac{(20/21)(1/4)}{1-(20/21)(3/4)} = \frac{5/21}{1-5/7} = \frac{5/21}{2/7} = \frac{5}{6}.

The duration-2 value is therefore $10{,}000(5/6)=\$8{,}333.33$, rounded to the nearest cent.

At age $x+1$, the death probability is $0.20$, and survival for one year has probability $0.80$. The value there is:

A_{x+1} = v(0.20)+v(0.80)A_{x+2} = \frac{4}{21}+\frac{16}{21}\cdot\frac{5}{6} = \frac{52}{63}.

At issue, the first-year death probability is $0.10$, and the probability of reaching age $x+1$ is $0.90$:

A_x = v(0.10)+v(0.90)A_{x+1} = \frac{2}{21}+\frac{6}{7}\cdot\frac{52}{63} = \frac{118}{147}.

Thus the single premium, equal to the issue value under the stated net assumptions, is $10{,}000(118/147)=\$8{,}027.21$, rounded to the nearest cent. The values check against the contract: the duration-2 value uses only mortality from age $x+2$ onward, while the issue value also includes the earlier, lower death probabilities.

Worked Example: Correctly Valuing a Four-Year Endowment

A hypothetical four-year endowment pays $25,000 at the end of the year of death, if death occurs within four years, or at the end of year 4 if the insured survives. Assume $i=5\%$ and $q=0.10$ in each year. Find the net single premium at issue.

Method. Add the present values of the four possible death benefits and the survival benefit. Here $v=20/21$, $p=0.90$, and $vp=6/7$. The discounted probability of death in year $j$ is $vq(vp)^{j-1}$, where $vq=2/21$. The discounted survival probability to year 4 is $(vp)^4$. Therefore the value per dollar of benefit is:

\frac{2}{21}\left[1+\frac{6}{7}+\left(\frac{6}{7}\right)^2+\left(\frac{6}{7}\right)^3\right] +\left(\frac{6}{7}\right)^4 = \frac{6098}{7203}.

For the fraction, the death-benefit sum in brackets is $1105/343$, so its contribution is $2210/7203$. The survival contribution is $1296/2401=3888/7203$; adding gives $6098/7203$. The single premium is:

25{,}000\left(\frac{6098}{7203}\right) = \$21{,}164.79.

The answer is rounded to the nearest cent. A common mistake is to use $0.10$ as the unconditional probability of death in every year, giving the incorrect factor $0.1v(1+v+v^2+v^3)+v^4$. That calculation fails to account for the need to survive to each later year; it even produces a factor greater than 1, which cannot be the value per dollar for this benefit and positive interest. The correction is to use death probabilities $0.1,\ 0.9(0.1),\ 0.9^2(0.1),\ 0.9^3(0.1)$, together with the separate survival probability $0.9^4$.

Worked Example: Endowment Value After One Year

A hypothetical three-year endowment pays $20,000 on death during the term or on survival to the end of year 3. Assume $i=4\%$, $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. Calculate the issue value and the prospective value at duration 1, conditional on survival to that duration.

Method. At issue, weight each possible payment by its unconditional probability. At duration 1, restart the calculation conditional on survival through year 1; only two years remain. The endowment benefit is paid at the end of year 3 whether the insured dies in that final year or survives, so those final-year outcomes together have probability equal to survival through the preceding year.

At issue, the probabilities of payment in years 1, 2, and 3 are $0.05$, $0.95(0.10)=0.095$, and $0.95(0.90)(0.20)=0.171$. The probability of survival to year 3 is $0.95(0.90)(0.80)=0.684$. The death and survival probabilities for the final payment date sum to $0.171+0.684=0.855$. With $v=1/1.04=25/26$, the issue value per dollar is:

0.05v+0.095v^2+0.855v^3 = 0.05\left(\frac{25}{26}\right) +0.095\left(\frac{25}{26}\right)^2 +0.855\left(\frac{25}{26}\right)^3 \approx 0.8960017.

The issue value is $20{,}000(0.8960017)\approx \$17{,}920.03$, rounded to the nearest cent.

At duration 1, the conditional probability of death in the next year is $0.10$. If the insured survives that year, the endowment pays at duration 2 years from the valuation time, whether death occurs in the final year or survival continues to maturity. The conditional probability of reaching that final payment date is $0.90$. Thus:

{}_1V_x = 20{,}000\left(0.10v+0.90v^2\right) = 20{,}000\left[ 0.10\left(\frac{25}{26}\right) +0.90\left(\frac{25}{26}\right)^2 \right] = \$18{,}565.09.

Both values are rounded to the nearest cent. The duration-1 value is conditional on survival to that duration; it does not include the year-1 death benefit or multiply future probabilities by the probability of surviving year 1 again.

Checks and Common Traps

Single-premium values are often straightforward once the future payment events are laid out. The main errors are usually about which events are covered, or about the time from the valuation date to payment. Before accepting a result, check the following:

Condition on survival at the valuation duration. At duration $k$, use mortality rates and survival probabilities from age $x+k$, not unconditional probabilities from issue.
Match the discount period to the payment date. A payment one year after the valuation date is discounted by $v$; a payment two years later is discounted by $v^2$.
Include maturity benefits only when promised. An endowment has a survival payment; a term policy does not.
Do not count both mutually exclusive outcomes as extra benefits. At an endowment’s final payment date, death and survival are separate ways to receive the same benefit, not two payments.
Use the right mortality model. A geometric-series whole-life formula requires the same annual mortality probability throughout the years being valued.

Key takeaway: With no future premiums, the prospective value is the conditional expected present value of future benefits. Identify the covered events, condition on survival to the valuation time, and discount each payment from that time.

Check Your Understanding

For each problem, use annual discrete benefits and the stated mortality and interest assumptions.

A hypothetical two-year endowment pays $10,000 on death during the term or survival to year 2. Assume $i=5\%$ and $q=0.20$ in each year. Find the net single premium at issue. (Answer: $9,161.00)
A hypothetical two-year term policy pays $20,000 at the end of the year of death. Assume $i=4\%$ and $q=0.20$ in each year. Find the net single premium. (Answer: $6,804.73)
A hypothetical whole-life policy pays $15,000 at the end of the year of death. Assume $i=4\%$ and a constant annual $q=0.20$ for life. Find its value at issue. (Answer: $12,500.00)
A hypothetical one-year term policy is valued immediately after duration 2, conditional on survival to then. Its benefit is $30,000, the next-year death probability is $0.15$, and $i=3\%$. Find the prospective value. (Answer: $4,368.93)
A hypothetical three-year endowment pays $12,000 on death during the term or survival to year 3. Assume $i=4\%$, $q_{x+1}=0.12$, and $q_{x+2}=0.25$. Find the value at duration 1, conditional on survival to that duration. (Answer: $11,147.93)

What You'll Learn