Cross-Check Contract Values with Independent Identities and Recursions

Why Independent Checks Matter

A life-contract calculation can be internally consistent and still be wrong: a cash flow may have been omitted, a payment may have been placed at the wrong time, or the reserve convention may have been misunderstood. A useful check is to value the contract in two different ways. First, compute present values directly from the possible cash flows. Then test selected reserves with the one-year recursion that follows from those same cash flows.

The two methods are not substitutes for one another. The cash-flow calculation checks the full contract value, while a recursion checks whether values at adjacent durations fit the annual transition. Both checks require the benefit and premium timing to be explicit. In particular, a recursion for a term-insurance year cannot be used unchanged in a year when a survival benefit is payable.

Definition: Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Write $q_{x+k}$ for the probability of death during the year from age $x+k$, conditional on being alive at its start, and $p_{x+k}=1-q_{x+k}$. A reserve ${}_kV_x$ measured immediately before the premium due at time $k$ is the present value at that time of future benefits minus the present value of future premiums, on the stated mortality and interest basis. The premium due at the valuation time is included among future premiums.

The Cash-Flow Identity as a Check

For a whole-life insurance paying 1 at the end of the year of death, $A_x$ is its expected present value. For a whole-life annuity-due paying 1 at the beginning of each year while the insured is alive, $\ddot{a}_x$ is its expected present value. These values use the same mortality and interest assumptions.

Theorem: For annual discrete whole-life insurance and an annuity-due on the same life, $A_x+d\ddot{a}_x=1$, where $d=1-v$. The identity assumes an insurance benefit at the end of the year of death and annuity payments at the beginning of each year of survival.

The identity provides an independent check: if either value is computed from a mortality table or a recurrence, the other can be checked against it. It follows by pairing the cash flows over each year of possible survival. Using ${}_kp_x$ for the probability of surviving $k$ years, the insurance value is $\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}$, and the annuity-due value is $\sum_{k=0}^{\infty}v^k{}_kp_x$. Since $q_{x+k}=1-p_{x+k}$, the combined coefficient in year $k$ is $d+vq_{x+k}=1-vp_{x+k}$. Therefore each term becomes the difference between the probability-weighted discount factors at times $k$ and $k+1$:

A_x+d\ddot{a}_x =\sum_{k=0}^{\infty}\left(v^k{}_kp_x-v^{k+1}{}_{k+1}p_x\right) =1.

The last equality is telescoping: the first term is 1, and the remaining term tends to 0 for a proper lifetime model with discounting. The identity is specific to these whole-life payment patterns; do not apply it without adjustment to a temporary contract or to a benefit payable on survival.

Worked Example: Checking Whole-Life Values and a Net Premium

A hypothetical whole-life policy pays $1,000 at the end of the year of death. Assume an effective annual interest rate of $i=0.10$ and a constant annual mortality probability $q_{x+k}=0.10$ at every duration, so $p_{x+k}=0.90$. Find $A_x$, $\ddot{a}_x$, and the level annual net premium payable in advance. Check the values independently.

Method. Under the stated assumptions, each additional year of survival and discounting multiplies the next term by $vp=(1/1.10)(0.90)=9/11$. Thus the whole-life insurance value is a geometric series:

A_x=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(10/11)(1/10)}{1-9/11} =\frac{1/11}{2/11} =\frac12.

The annuity-due value is another geometric series:

\ddot{a}_x=\sum_{k=0}^{\infty}(vp)^k =\frac{1}{1-9/11} =\frac{11}{2}=5.5.

The level net premium $P$ follows by equivalence: the expected present value of premiums equals that of benefits. Therefore $P\ddot{a}_x=1000A_x$, so $P=1000(0.5)/5.5=\$90.9090\ldots$, or $90.91 to the nearest cent.

Independent check. Here $d=1-v=1/11$. The identity gives $A_x+d\ddot{a}_x=0.5+(1/11)(5.5)=0.5+0.5=1$. The premium also checks by substitution: $90.9090\ldots\times5.5=500=1000\times0.5$. This check would fail if the insurance benefit were incorrectly treated as payable at the beginning of the year.

One-Year Recursions and Their Conditions

A one-year recursion relates the reserve immediately before a premium to the reserve one year later. For a level annual premium $P$ and a death benefit $B$ paid at the end of the year of death, if no benefit is paid on survival during that year, the recursion is:

({}_kV_x+P)(1+i) =q_{x+k}B+p_{x+k}{}_{k+1}V_x.

The timing explains the equation. At time $k$, the premium is received along with the reserve, and the amount accumulates for one year. At year-end, the death branch pays $B$; the survival branch carries the reserve for the next duration. Taking the conditional expected value of these year-end amounts gives the right-hand side. This form is appropriate, for example, for a term-insurance year with no survival payment.

If a survival benefit $S$ is paid at year-end, the survival branch must include it. When ${}_{k+1}V_x$ is measured after that benefit is paid, the appropriate equation is:

({}_kV_x+P)(1+i) =q_{x+k}B+p_{x+k}\left(S+{}_{k+1}V_x\right).

The form of the recursion depends on the contract cash flows and the reserve convention. State both before calculating. In particular, a maturity benefit is not a death benefit: it belongs in the survival branch, even if it is paid at the same year-end.

Worked Example: Cross-Checking a Two-Year Endowment

A hypothetical two-year endowment pays $2,000 at death during the term or at time 2 on survival. The annual death probabilities are $q_x=0.10$ and $q_{x+1}=0.20$, with $p_x=0.90$ and $p_{x+1}=0.80$. The effective annual interest rate is $i=0.10$. Level premiums are payable at times 0 and 1, provided the insured is alive. Find the net premium and the duration-1 reserve immediately before the premium. Check the recursion in both years.

Method. First value the insurance benefits directly. The probabilities of death in years 1 and 2 are $0.10$ and $0.90(0.20)=0.18$; the probability of surviving to maturity is $0.90(0.80)=0.72$. With $v=10/11$, the benefit present value is:

2000\left(0.10v+0.90(0.20)v^2+0.90(0.80)v^2\right) =2000\left(\frac{11+18+72}{121}\right) =2000\left(\frac{101}{121}\right) =\$1,669.42.

The expected present value of the premium pattern per dollar of premium is $1+0.90v=1+9/11=20/11$. Equating premium and benefit present values gives:

P\left(\frac{20}{11}\right)=2000\left(\frac{101}{121}\right), \qquad P=\frac{202000}{220}=\$918.18

to the nearest cent. At duration 1, the $2,000 benefit is paid at the end of the next year whether death occurs then or the insured survives to maturity. The future benefit value is therefore $2000v=\$1,818.18$. The premium due immediately at time 1 is included in the prospective value, so ${}_1V_x=2000v-P=\$900.00$. At duration 2, after maturity, ${}_2V_x=0$.

Recursion check for year 1. This is the final year, so the survival benefit is $S=\$2,000$. The left side is $(900+918.1818\ldots)(1.10)=\$2,000$. The right side is $0.20(2,000)+0.80(2,000+0)=\$2,000$.

Recursion check for year 0. No benefit is paid on survival at time 1, so use the no-survival-benefit form. The left side is $(0+918.1818\ldots)(1.10)=\$1,010$. The right side is $0.10(2,000)+0.90(900)=\$1,010$. Both checks agree with the direct valuation.

Cross-Checking a Term-Insurance Premium and Reserve

For a term policy, the direct cash-flow calculation and the recursion provide different checks on the same premium and reserves. The direct calculation sums the probability-weighted present values of death benefits and of premiums paid while in force. The recursion then checks that the reserve transition agrees with the annual death and survival probabilities.

Worked Example: Two-Year Term Insurance with Unequal Mortality

A hypothetical two-year term policy pays $1,000 at the end of the year of death and nothing on survival to time 2. Assume $i=0.10$, $q_x=0.10$, $q_{x+1}=0.20$, and level premiums payable at times 0 and 1 while the insured is alive. Calculate the net premium and duration-1 pre-premium reserve, then check both yearly recursions.

Method. The probability of a first-year death is $0.10$, and the probability of a second-year death is $0.90(0.20)=0.18$. With $v=10/11$, the benefit present value is:

1000\left(0.10v+0.90(0.20)v^2\right) =1000\left(\frac{11+18}{121}\right) =1000\left(\frac{29}{121}\right) =\$239.67.

The premium annuity-due factor is $1+0.90v=20/11$. Equivalence gives $P=(1000\times29/121)/(20/11)=\$131.8181\ldots$, or $131.82 to the nearest cent. For the duration-1 reserve, the remaining one-year death-benefit value is $1000(0.20)v=\$181.8181\ldots$. Subtracting the premium due at time 1 gives ${}_1V_x=181.8181\ldots-131.8181\ldots=\$50.00$. The terminal reserve is ${}_2V_x=0$.

Recursion check for year 0. The left side is $(0+131.8181\ldots)(1.10)=\$145$. The right side is $0.10(1,000)+0.90(50)=\$145$.

Recursion check for year 1. The left side is $(50+131.8181\ldots)(1.10)=\$200$. The right side is $0.20(1,000)+0.80(0)=\$200$. There is no survival benefit in either year, so the term-insurance recursion applies in both. The direct premium valuation and both local checks agree.

Common Mistakes and a Practical Checking Sequence

The most consequential error in an integrated calculation is often not arithmetic but a missing or mistimed cash flow. For example, in the endowment’s final year, omitting the survival benefit produces a right side of only $0.20(2,000)=\$400$, while the correctly accumulated amount is $\$2,000$. The omitted survival branch is $0.80(2,000)=\$1,600$. The correction is to include that maturity payment in the survival branch; it is not to alter the death probability or the interest rate.

Do not use a term-insurance recursion for a maturity year. Add any survival benefit due at year-end to the survival branch.
Do not confuse an annual probability with a multi-year probability. A second-year death probability unconditional from issue includes survival through the first year.
Do not omit a premium due at the reserve date. Under the pre-premium convention used here, it is included in the prospective premium value.
Do not mix payment timing in an identity. The whole-life identity above pairs an end-of-year death benefit with an annuity-due.
Do not use rounded intermediate values for the final check. Retain adequate precision, then round reported dollar amounts to cents.

A disciplined check can be carried out in four steps: list the payment timing and survival conditions; value benefits and premiums separately; compute the selected prospective reserve using the same assumptions; and test each relevant year with the matching recursion. If the two methods disagree, inspect the boundary year first: maturity, the final premium, and the timing of a benefit are frequent sources of error. This verification approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, state the cash-flow timing before calculating. Dollar answers are rounded to the nearest cent where needed.

A hypothetical whole-life policy has $i=0.25$ and constant annual mortality $q_{x+k}=0.20$. Find $A_x$, $\ddot{a}_x$, and verify $A_x+d\ddot{a}_x=1$. (Answer: $A_x=4/9$, $\ddot{a}_x=25/9$, and $d=0.20$, so the identity equals 1.)
A two-year endowment pays $1,000 at death during the term or at time 2 on survival. Let $i=0.25$, $q_x=0.20$, and $q_{x+1}=0.10$; level premiums are payable at times 0 and 1 while in force. Find the net premium and the duration-1 pre-premium reserve. (Answer: $P=\$409.76$, ${}_1V_x=\$390.24$.)
A two-year term policy pays $2,000 at the end of the year of death and nothing on survival. Assume $i=0.05$, $q_x=0.05$, and $q_{x+1}=0.15$, with premiums payable at times 0 and 1 while in force. Find the net premium and duration-1 pre-premium reserve. (Answer: $P=\$185.71$, ${}_1V_x=\$100.00$.)
A one-year contract has no premium, $i=0.10$, $q_x=0.20$, and $p_x=0.80$. It pays $1,000 at year-end on death and $500 at year-end on survival. With zero reserve after the year, find the initial prospective value and write the correct one-year recursion. (Answer: ${}_0V_x=\$545.45$; \({}_0V_x(1.10)=0.20(1,000)+0.80(500)=\$600.)

What You'll Learn