Bounds and Checks for Discrete Insurance APVs

Why Check an Insurance APV?

A discrete life insurance APV is a sum of discounted benefits weighted by death probabilities. That sum can be evaluated correctly only if the probability in each term matches the policy year, and the discount factor matches the benefit payment time. A numerical answer may look plausible while containing an off-by-one error, an incorrect probability, or an inconsistent interest rate.

Simple bounds and direction checks provide an independent audit. They do not replace the APV calculation, but they can establish that a result is impossible before it is used in a premium or valuation calculation. The checks below apply to level-benefit discrete term and whole-life insurance, with benefit paid at the end of the policy year of death.

Definition: Let $K_x$ be the curtate future lifetime of a life aged $x$. Under the end-of-year-of-death convention, a death in the first policy year corresponds to $K_x=0$, and the benefit is paid at time $K_x+1$. For a level benefit $B$, annual effective interest rate $i$, and $v=1/(1+i)$, the APV is the expected present value of the benefit. A term policy pays only if death occurs within its stated term; a whole-life policy pays on death at any future duration.

Probability Totals Come First

The probabilities in the APV sum are usually unconditional probabilities of dying in each policy year, not the conditional probabilities $q_{x+k}$ viewed separately. For a term of $n$ years, the probability of death in policy year $k+1$, for $k=0,\ldots,n-1$, is ${}_kp_xq_{x+k}$. These probabilities sum to the probability of dying during the term:

{}_nq_x=\sum_{k=0}^{n-1}{}_kp_xq_{x+k}=1-{}_np_x.

The probability remaining after the term, ${}_np_x$, completes the total to 1. This is a useful audit: if the yearly death probabilities and the probability of surviving the term do not add to 1, at least one probability or survival factor has been mishandled.

Formula: For level benefit $B$, a discrete $n$-year term insurance has $$ \text{APV}=B A_{x:\overline{n}|} =B\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}. $$ A whole-life insurance has $$ \text{APV}=B A_x =B\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}, $$ when the sum is defined. In a finite mortality table, the sum ends at the last possible year of death.

Bounds for a Term Insurance APV

Suppose $i\geq 0$, so $0<v\leq 1$. In an $n$-year term, the benefit is paid at one of the times $1,2,\ldots,n$, if death occurs during the term. Thus its discount factor is between $v^n$ and $v$. Multiplying those bounds by the probability of death during the term gives a quick range for the APV.

Bv^n\,{}_nq_x \leq B A_{x:\overline{n}|} \leq Bv\,{}_nq_x.

These bounds are deliberately broad: they treat every covered death as though it occurred at the latest or earliest possible payment time. They are still useful for catching errors. The APV must also be nonnegative and no greater than $B\,{}_nq_x$, because a discount factor cannot exceed 1 when $i\geq 0$.

The duration-specific range requires nonnegative interest. If $i<0$, then $v>1$, and the ordering of the discount factors used in this bound does not hold. The APV remains an expected present value, but these particular inequalities should not be applied without re-deriving the bounds for the rate in question.

Worked Example: Bounding a Three-Year Term APV

A hypothetical insurer offers a three-year discrete term policy with a benefit of $B=\$10{,}000$, payable at the end of the policy year of death. The annual effective interest rate is $5\%$. Conditional death probabilities for the three policy years are $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Calculate the APV and check it against the probability total and term bounds.

Method. First convert the conditional probabilities to unconditional policy-year probabilities. Then calculate the APV with the corresponding end-of-year discount factors, and compare it with the bounds.

The survival probability to the start of year 1 is 1, so the probability of death in year 1 is $0.10$. Survival to the start of year 2 is $1-0.10=0.90$, giving a year-2 death probability of $0.90(0.20)=0.18$. Survival to the start of year 3 is $0.90(1-0.20)=0.72$, giving a year-3 death probability of $0.72(0.25)=0.18$. The total probability of death during the term is $0.10+0.18+0.18=0.46$. The probability of surviving the term is $0.72(0.75)=0.54$, and $0.46+0.54=1.00$.

At $i=0.05$, $v=1/1.05=0.952381$, $v^2=0.907029$, and $v^3=0.863838$. The three discounted expected benefit amounts are:

\begin{aligned} \text{Year 1: }&\$10{,}000(0.952381)(0.10)=\$952.38,\\ \text{Year 2: }&\$10{,}000(0.907029)(0.18)=\$1{,}632.65,\\ \text{Year 3: }&\$10{,}000(0.863838)(0.18)=\$1{,}554.91. \end{aligned}

Adding the unrounded amounts gives an APV of $\$4{,}139.94$. The lower bound is $\$10{,}000(0.863838)(0.46)=\$3{,}973.65$; the upper bound is $\$10{,}000(0.952381)(0.46)=\$4{,}380.95$. The calculated value lies between these bounds, is nonnegative, and is below the undiscounted expected benefit of $\$10{,}000(0.46)=\$4{,}600$. These checks support, but do not by themselves prove, that the calculation is correct.

Common mistake avoided. Adding $0.10+0.20+0.25$ treats the conditional death probabilities as unconditional. The correct total probability is $0.46$, not $0.55$, because a life must survive to each policy year before its conditional death probability can apply.

Bounds for Whole-Life Insurance

For a whole-life policy, death can occur in any future policy year. The APV is nonnegative when the benefit and discount factors are nonnegative. If $i\geq 0$, each payment is discounted by at most $v$, because the earliest possible payment is at the end of year 1. Therefore,

0\leq B A_x \leq Bv\sum_{k=0}^{\infty}{}_kp_xq_{x+k} \leq Bv.

The sum of the unconditional probabilities of death in all future years is at most 1. It equals 1 if death eventually occurs with probability 1; in that case the bound can be sharpened to $BA_x\leq Bv$. Even the simpler bound $BA_x\leq B$ is a useful check for nonnegative interest. For a finite mortality table, the same reasoning applies to its finite set of possible death years, provided those probabilities are handled consistently.

A particularly strong check is available when the mortality model and interest rate make the whole-life APV easy to evaluate independently. Comparing a finite sum, a recurrence, or a table calculation with that separate value can reveal an indexing or truncation error. The bound is a screening check, not a substitute for checking the terms themselves.

Worked Example: Checking a Whole-Life APV Under Constant Mortality

A hypothetical whole-life policy pays $\$25{,}000$ at the end of the year of death. Assume a constant conditional death probability $q=0.20$ each year, so $p=0.80$, and an annual effective interest rate of $5\%$. Calculate the APV and check the result against whole-life bounds.

Method. Under this model, the probability of death in year $k+1$ is $p^kq$. The APV factor is a geometric series with first term $vq$ and common ratio $vp$:

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp}, \qquad |vp|<1.

Here $v=1/1.05=20/21$, so $vq=(20/21)(1/5)=4/21$ and $vp=(20/21)(4/5)=16/21$. Therefore, $1-vp=1-16/21=5/21$, and $A_x=(4/21)/(5/21)=0.80$. The APV is $\$25{,}000(0.80)=\$20{,}000.00$.

The probability of eventual death is $\sum_{k=0}^{\infty}p^kq=q/(1-p)=0.20/0.20=1$. The whole-life bound is therefore $0\leq\text{APV}\leq Bv=\$25{,}000(20/21)=\$23{,}809.52$; the weaker bound $B=\$25{,}000$ also holds. The calculated $\$20{,}000.00$ satisfies both checks.

Common mistake avoided. The first payment is at the end of year 1, not immediately. Starting the series with $q$ instead of $vq$ omits one year of discounting and would produce an incorrect APV.

Direction Checks for Duration and Interest

For the same benefit, mortality assumptions, and interest rate, extending a term cannot reduce its APV: the longer contract includes all the benefits covered by the shorter contract and may cover additional deaths. A term APV also cannot exceed the corresponding whole-life APV, because whole life covers every death that the term covers, as well as any later death.

For fixed mortality and coverage, increasing a nonnegative interest rate reduces or leaves unchanged each discount factor $v^{k+1}$. Since the probabilities and benefit are nonnegative, the APV cannot increase. These direction checks concern the effect of changing one feature while holding the others fixed; they do not assert a particular direction if mortality, benefit design, or other assumptions change at the same time.

Check the probability inputs.
For each policy year, multiply the conditional death probability by the probability of surviving to that year. Confirm that covered death probabilities sum to ${}_nq_x$.

Check the timing.
Under the end-of-year-of-death convention, year $k+1$ uses discount factor $v^{k+1}$, not $v^k$.

Check bounds and direction.
For nonnegative interest, verify nonnegativity, the applicable benefit-based upper bound, and the expected direction under a change in duration or interest rate.

Worked Example: Checking Interest and Duration Effects

Use a hypothetical three-year term policy with benefit $B=\$5{,}000$, conditional death probabilities $0.10$, $0.20$, and $0.25$, and end-of-year-of-death payment. Compare its APVs at $4\%$ and $8\%$, and compare the three-year value at $4\%$ with the two-year value at $4\%$.

Method. The unconditional probabilities are $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(0.25)=0.18$. Their sum is $0.46$. At $4\%$, $v=1/1.04=0.961538$, $v^2=0.924556$, and $v^3=0.888996$. Thus the three-year APV is $\$5{,}000[(0.10)(0.961538)+(0.18)(0.924556)+(0.18)(0.888996)]=\$2{,}112.97$, rounded to the nearest cent. The two-year APV omits only the third-year term: $\$5{,}000[(0.10)(0.961538)+(0.18)(0.924556)]=\$1{,}312.87$.

At $8\%$, $v=1/1.08=0.925926$, $v^2=0.857339$, and $v^3=0.793832$. The three-year APV is $\$5{,}000[(0.10)(0.925926)+(0.18)(0.857339)+(0.18)(0.793832)]=\$1{,}949.02$, rounded to the nearest cent. The three-year value at $4\%$ exceeds the two-year value at $4\%$, as expected; it also exceeds the three-year value at $8\%$. These comparisons verify the expected duration and interest directions.

Common mistake avoided. A higher rate does not make the probability of death smaller in this calculation; it makes the present value of each future benefit smaller. The mortality probabilities stay fixed when comparing the two interest rates.

How to Use These Checks

A useful audit moves from the probability model to the money calculation. First verify survival to each policy year and the resulting death probabilities. Next verify that the benefit timing agrees with the exponent on $v$. Then check nonnegativity and the relevant benefit-based bounds. Finally, compare results across related contracts or interest rates.

No single check detects every error. For example, an off-by-one discounting error can still produce a positive value below $B$. A wrong probability can also produce a value within a broad range by coincidence. The strongest audit combines independent checks and, when possible, compares the APV with a separately calculated result.

Key takeaway: For discrete insurance APVs, check that unconditional death probabilities are correctly formed and total to the appropriate coverage probability. With nonnegative interest, a term APV is bounded using the covered death probability and the earliest and latest payment times; a whole-life APV is nonnegative and cannot exceed $Bv$. Extending term coverage cannot lower the APV, while increasing interest cannot raise it when all other assumptions are fixed.

Check Your Understanding

For each problem, apply the probability, bound, or direction check that fits the stated model.

A two-year term pays $\$12{,}000$ at the end of the year of death. The conditional death probabilities are $0.05$ in year 1 and $0.10$ in year 2, and $i=0$. Find the APV and confirm the probability of surviving the term. (Answer: APV $=\$1{,}740.00$; survival probability $=0.855$.)
A whole-life policy pays $\$30{,}000$ at the end of the year of death. Assume constant $q=0.10$, $p=0.90$, and $i=0.04$. Find the APV and the upper bound $Bv$. (Answer: APV $=\$21{,}428.57$; $Bv=\$28{,}846.15$.)
A three-year term has $B=\$10{,}000$, $i=0.05$, and conditional death probabilities $0.10$, $0.20$, and $0.25$. Use the probability of death during the term to find the lower and upper duration-specific bounds. (Answer: $\$3{,}973.65\leq\text{APV}\leq\$4{,}380.95$.)
For a two-year term with $B=\$5{,}000$, conditional death probabilities $0.10$ and $0.20$, and $i=0.04$, calculate the APV. State whether extending coverage to three years using a third-year conditional death probability of $0.25$ can reduce it. (Answer: two-year APV $=\$1{,}312.87$; extension cannot reduce the APV.)
A candidate adds conditional death probabilities $0.10$, $0.20$, and $0.25$ to obtain a three-year death probability. What is the correct three-year death probability, and what probability was omitted from the candidate’s calculation? (Answer: $0.46$; the candidate omitted survival weighting, so the correct unconditional probabilities are $0.10$, $0.18$, and $0.18$.)

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