Truncating Whole-Life Calculations With a Residual Tail

Why a Finite Table Needs a Residual Tail

A whole-life insurance benefit is payable whenever death occurs, so its actuarial present value (APV) includes contributions from every possible death year. A supplied mortality table, however, may give death probabilities or life-table counts only for a finite number of years. Summing the terms that appear in the table gives the observed portion of the APV, not automatically the whole-life value. The contribution from deaths after the table ends is the residual tail.

The key is to keep two questions separate: how much value comes from the years actually shown, and what can be justified about the remaining lives? If the table does not specify the later mortality pattern, a point estimate requires an additional assumption. Without one, report bounds supported by the information available.

Definition: Consider a unit benefit payable at the end of the policy year of death. Let $i$ be the effective annual interest rate, $v=1/(1+i)$, and assume $i>0$. At age $x$, the whole-life insurance APV is $$ A_x=\sum_{k=0}^{\infty} v^{k+1}\,{}_kp_x\,q_{x+k}, $$ where ${}_kp_x$ is the probability of surviving $k$ years and $q_{x+k}$ is the one-year probability of death between ages $x+k$ and $x+k+1$, conditional on being alive at age $x+k$.

If the table supplies the first $m$ policy-year death probabilities, the terms for those years run from $k=0$ through $k=m-1$. The probability of surviving all $m$ years is ${}_mp_x$. Conditional on reaching age $x+m$, let $A_{x+m}^{\mathrm{tail}}$ denote the APV of a unit whole-life benefit measured from that age. The decomposition is

A_x= \underbrace{\sum_{k=0}^{m-1}v^{k+1}\,{}_kp_x\,q_{x+k}}_{\text{observed portion}} + \underbrace{v^m\,{}_mp_x\,A_{x+m}^{\mathrm{tail}}}_{\text{residual tail}}.

The factor $v^m$ moves the value at age $x+m$ back to the original valuation time. The factor ${}_mp_x$ accounts for the chance that the insured is still alive to enter the unobserved period. Both factors are needed.

Calculating the Observed Portion

The observed terms use the same death-year partition as a whole-life APV from age-specific mortality rates or life-table counts. For each year, multiply the probability of surviving to its start by the conditional probability of dying during that year, then discount the benefit to the end of that year.

Worked Example: Observed Years and a Two-Year Tail Bound

A hypothetical policy pays 1 at the end of the year of death. The effective annual interest rate is $i=25\%$, so $v=1/1.25=0.8$. The one-year death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. All insureds who survive the first two years are known to die within the next two years, but the split between those two later years is not given. Find the observed APV and bounds for the whole-life APV.

Method. First calculate the first two death-year contributions. Then bound the conditional tail using the earliest and latest possible death years in the two-year residual period.

The first-year death probability is $0.10$, so the first-year contribution is $0.10(0.8)=0.08$.

Survival through year 1 is $1-0.10=0.90$. Thus the unconditional probability of death in year 2 is $0.90(0.20)=0.18$, and its contribution is $0.18(0.8^2)=0.18(0.64)=0.1152$.

The observed value is therefore $0.08+0.1152=0.1952$. The surviving proportion at the end of year 2 is ${}_2p_x=0.90(1-0.20)=0.90(0.80)=0.72$.

Conditional on surviving to the start of the residual period, death occurs in either policy year 3 or policy year 4. Measured from the original valuation time, those payments are discounted by $v^3$ and $v^4$. Since $0<v<1$, the conditional tail value is at least $v^4=0.4096$ and at most $v^3=0.512$. The residual contribution at the original valuation time is consequently between $0.72(0.4096)=0.294912$ and $0.72(0.512)=0.36864$.

Adding the observed value gives $0.1952+0.294912=0.490112$ and $0.1952+0.36864=0.56384$. Thus the whole-life APV is between $0.490112$ and $0.56384$.

Check and trap avoided. The observed death probabilities sum to $0.10+0.18=0.28$, leaving $0.72$ alive at the cutoff. The residual is bounded using that survivor proportion, not by assigning the original cohort a second set of probabilities. The range is a bound, not a point estimate, because the timing of those $0.72$ residual deaths is unknown.

Justifying the Residual Bounds

Suppose $i>0$, so $0<v<1$, and assume every life surviving to the cutoff dies within the next $r$ policy years. Conditional on survival to that cutoff, let $w_j$ be the probability of death in the $j$th subsequent year, for $j=1,\ldots,r$. These probabilities are nonnegative and sum to 1. The conditional APV of the tail, measured at the cutoff, is

A_{x+m}^{\mathrm{tail}}=\sum_{j=1}^{r} w_jv^j.

Because $0<v<1$, the smallest discount factor among these years is $v^r$ and the largest is $v$. Multiplying those inequalities by each $w_j$, adding, and using $\sum_{j=1}^{r}w_j=1$ gives

v^r\leq \sum_{j=1}^{r}w_jv^j\leq v.

Discounting this conditional value back $m$ years and multiplying by the probability of surviving to the cutoff gives the residual bounds

{}_mp_xv^m v^r \leq v^m{}_mp_xA_{x+m}^{\mathrm{tail}} \leq {}_mp_xv^m v.

The bounds depend on the positive-interest assumption. If $i=0$, every discount factor is 1, so death guaranteed within the residual period gives a conditional tail value of 1. If $i<0$, then $v>1$, and the discount factors increase rather than decrease with time; for death within $r$ years the corresponding conditional bounds are $v\leq A_{x+m}^{\mathrm{tail}}\leq v^r$. Do not use the positive-interest bounds in that case.

Find survival to each observed year.
Use the supplied conditional death probabilities or life-table counts to obtain the probability alive at the start of each policy year.

Sum the observed contributions.
For each supplied year, multiply its unconditional death probability by the end-of-year discount factor.

Identify the residual information.
Record the survivor probability at the cutoff and state whether the remaining death timing is bounded, specified, or modeled.

Value or bound the tail.
Apply a justified timing bound, or state a continuation mortality assumption and calculate the resulting conditional APV.

Combine and check.
Add the observed value and residual contribution. Check that death probabilities and survivor probabilities are consistent with the cohort.

Estimating a Tail Under an Explicit Continuation Model

When no later table entries are supplied, an estimate can be obtained by specifying a mortality model for the residual period. For example, suppose the conditional probability of death in each future year is the same constant $q$, independently in the sense that the same conditional probability applies at every year survived. The probability of death in the $j$th year after the cutoff is then $(1-q)^{j-1}q$. For $0<v<1$, the conditional tail APV is a convergent geometric series:

A_{x+m}^{\mathrm{tail}} =\sum_{j=1}^{\infty}v^j(1-q)^{j-1}q =\frac{vq}{1-v(1-q)}.

The series sums because $0\leq v(1-q)<1$. This is an estimate for the original whole-life value only if constant mortality is a reasonable stated assumption for the unobserved ages; it is not implied merely by having a finite table.

Worked Example: Geometric Tail Estimate

A hypothetical unit-benefit policy has effective annual interest $i=25\%$, hence $v=0.8$. The first two conditional death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. After the second year, assume as a model that the conditional death probability is $q=0.20$ in every year. Estimate $A_x$.

Method. Calculate the two observed terms, then value the residual from age $x+2$ using the geometric-series formula.

As in the first example, the observed value is $0.10(0.8)+[0.90(0.20)](0.8^2)=0.08+0.1152=0.1952$. The probability of surviving two years is $0.90(0.80)=0.72$.

The conditional tail APV at age $x+2$ is

\frac{vq}{1-v(1-q)} = \frac{0.8(0.20)}{1-0.8(0.80)} = \frac{0.16}{1-0.64} = \frac{0.16}{0.36} = \frac{4}{9}.

The tail contribution at age $x$ is $0.72(0.8^2)(4/9)=0.72(0.64)(4/9)=0.4608(4/9)=0.2048$. Therefore the estimated whole-life APV is $0.1952+0.2048=0.4000$.

Check and trap avoided. The conditional tail value $4/9$ is measured at age $x+2$, not at age $x$. It must be multiplied by both the probability $0.72$ of reaching that age and the two-year discount factor $0.64$.

Using Life-Table Counts and Reporting Uncertainty

If a table gives $\ell_x$, the number alive at age $x$, and later counts, the probability of death in a year can be calculated as a fraction of the original cohort. For example, the unconditional probability of death between ages $x+k$ and $x+k+1$, for a life initially selected at age $x$, is $(\ell_{x+k}-\ell_{x+k+1})/\ell_x$, when the counts refer to the same life-table basis. The fraction still alive at the cutoff is $\ell_{x+m}/\ell_x$. These quantities should reconcile: observed death fractions plus the survivor fraction equal 1.

Worked Example: Counts, a Guaranteed Residual Period, and Bounds

A hypothetical life table starts with $\ell_x=1{,}000$ lives and gives $\ell_{x+1}=900$, $\ell_{x+2}=720$, and $\ell_{x+3}=540$. A unit benefit is paid at the end of the death year. The effective annual interest rate is $i=1/19$, so $v=0.95$. Assume all lives remaining at age $x+3$ die within the next two years, but their split between those years is unknown. Find the observed APV and bounds for the whole-life APV.

Method. Convert the annual count decreases to unconditional death probabilities by dividing by the original $1{,}000$, calculate their discounted contributions, then bound the residual using the two possible later payment years.

Deaths in year 1 number $1{,}000-900=100$, giving probability $100/1{,}000=0.10$. Deaths in year 2 number $900-720=180$, giving probability $0.18$. Deaths in year 3 number $720-540=180$, also giving probability $0.18$. The observed value is

$$ 0.10(0.95)+0.18(0.95^2)+0.18(0.95^3) =0.095+0.16245+0.1543275 =0.4117775. $$

The fraction surviving to age $x+3$ is $540/1{,}000=0.54$, consistent with $0.10+0.18+0.18+0.54=1$. The residual deaths occur in years 4 or 5, whose original-time discount factors are $0.95^4$ and $0.95^5$. Equivalently, the contribution is between $0.54(0.95^3)(0.95^2)$ and $0.54(0.95^3)(0.95)$.

Since $0.95^3=0.857375$, $0.95^2=0.9025$, the lower residual bound is $0.54(0.857375)(0.9025)=0.41784170625$. The upper residual bound is $0.54(0.857375)(0.95)=0.439833375$.

Adding the observed value gives a whole-life APV between $0.4117775+0.41784170625=0.82961920625$ and $0.4117775+0.439833375=0.851610875$. The result is a range because the counts do not reveal which of the two residual years contains each death.

Check and trap avoided. The residual probability is $0.54$, the proportion alive at the cutoff. It cannot be assigned as a death probability in each of the two later years: the probabilities of death in those years must together account for, and cannot exceed, the $0.54$ remaining lives.

Common Errors and Practical Checks

Truncation is useful when a table is incomplete, but it makes the assumptions behind the answer especially visible. A reported point value should identify the model used for the residual tail. If only a range is justified, report the range rather than silently choosing a convenient residual mortality pattern.

Discounting from the wrong time. A conditional APV calculated at the cutoff must be multiplied by $v^m$ to return to the original valuation time.
Omitting survival to the cutoff. Only the proportion ${}_mp_x$ that reaches the residual period can generate tail benefits.
Confusing conditional and unconditional death probabilities. A one-year $q_{x+k}$ is conditional on being alive at that year’s start. Multiply by the probability of reaching that start before using it as a contribution from the original cohort.
Using a bound without its rate assumption. The simple earliest-to-latest discount-factor bounds above require $i>0$; they do not apply unchanged for a negative rate.
Reporting a model-based estimate as table information. A constant-$q$ continuation is an additional assumption, not a fact established by the observed entries.

A final reconciliation is often decisive: the observed unconditional death probabilities, the residual unconditional death probability, and any probability still alive after a guaranteed terminal period should sum to 1. This check catches inconsistent cohort splits before they contaminate the APV.

Key takeaway: A truncated whole-life calculation equals the observed death-year sum plus a residual tail. Value that tail only with information or assumptions that are stated explicitly, and carry both the cutoff survival probability and the discount back to the original age.

Check Your Understanding

For each problem, use a unit benefit payable at the end of the death year. Assume positive effective annual interest unless stated otherwise.

A hypothetical policy has $i=25\%$, $q_x=0.20$, and $q_{x+1}=0.10$. All lives surviving year 2 die in year 3. Find the observed value and whole-life APV. (Answer: Observed value $0.2112$; whole-life APV $0.57984$.)
A hypothetical table gives $q_x=0.15$, and $i=1/9$, so $v=0.90$. Death for every life surviving year 1 is guaranteed within the next three years, but its timing is unknown. Find bounds for the whole-life APV. (Answer: Observed value $0.135$; whole-life APV is between $0.692685$ and $0.8235$.)
A hypothetical cohort has $\ell_x=500$, $\ell_{x+1}=450$, and $\ell_{x+2}=300$. Let $i=25\%$. After age $x+2$, assume a constant conditional annual death probability $q=0.25$. Estimate the whole-life APV. (Answer: Observed value $0.272$; residual $0.192$; estimated whole-life APV $0.464$.)
A hypothetical policy has $i=1/19$, $q_x=0.20$, and $q_{x+1}=0.375$. The surviving fraction after year 2 is $0.50$. Conditional on reaching year 3, the probability of death in year 3 is $0.40$, and all others die in year 4. Find the observed value, residual contribution, and whole-life APV. (Answer: Observed value $0.46075$; residual $0.415826875$; whole-life APV $0.876576875$.)

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