Cross-Checking Multi-Step Insurance-Annuity Calculations

Why Cross-Checking Matters

A multi-stage life contract can be valued by listing each possible benefit payment, discounting it, and weighting it by the probability it is paid. That direct calculation is essential, but it is not the only way to check a result. Under consistent assumptions, insurance and annuity values are linked: a death benefit during a period and an annuity payable while alive over that same period satisfy a complement identity. A second calculation using that identity can expose a missed survival factor, an incorrect time of payment, or a boundary error.

The checks in this tutorial use annual effective interest and annual benefit payments. They are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The contracts are hypothetical, and benefits are assumed to have unit amount unless stated otherwise.

Definition: For a life aged $x$, let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let ${}_kp_x$ be the probability of surviving $k$ years from age $x$, and let ${}_kq_x=1-{}_kp_x$. The $n$-year term insurance $A^1_{x:\overline{n}|}$ pays 1 at the end of the year of death if death occurs within $n$ years. The $n$-year temporary life annuity-due $\ddot{a}_{x:\overline{n}|}$ pays 1 at times $0,1,\ldots,n-1$, provided the life is alive at the payment time. The pure endowment $v^n{}_np_x$ pays 1 at time $n$ if the life survives $n$ years. An $n$-year endowment insurance pays 1 at the end of the year of death if death occurs within $n$ years, or 1 at time $n$ if the life survives to then.

The Complement Identity and Its Reason

With these payment timings, the finite-term complement identity is

A^1_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|} =1-v^n{}_np_x, \qquad d=1-v.

The identity compares two different expected present values. The term insurance pays on death within the term; the annuity-due payments occur at the beginnings of the years while the life is alive. The right-hand side is the complement of the discounted pure endowment.

To see why the identity holds, consider year $k$, for $k=1,\ldots,n$. The probability of death during that year is ${}_{k-1}p_xq_{x+k-1}$. The term insurance payment for that death is made at time $k$, while the corresponding annuity payment is made at time $k-1$ if the life is alive then. Their contributions to the left side for that year are

v^k{}_{k-1}p_xq_{x+k-1} +d\,v^{k-1}{}_{k-1}p_x = v^{k-1}{}_{k-1}p_x\bigl(vq_{x+k-1}+d\bigr).

Since $q_{x+k-1}=1-p_{x+k-1}$ and $d=1-v$, the expression in parentheses is $1-vp_{x+k-1}$. Thus the year-$k$ contribution equals

v^{k-1}{}_{k-1}p_x-v^k{}_kp_x.

Summing over $k=1,\ldots,n$ makes the intermediate terms cancel: the first term is 1 and the final subtracted term is $v^n{}_np_x$. This proves the identity. The same reasoning is useful as a practical check: every insurance death payment must use the probability of death in that year, and every annuity payment must use survival to its payment time.

Theorem: An $n$-year endowment insurance is the sum of an $n$-year term insurance and a pure endowment with the same benefit amount. Consequently, $$ A_{x:\overline{n}|}=A^1_{x:\overline{n}|}+v^n{}_np_x, \qquad A_{x:\overline{n}|}+d\ddot{a}_{x:\overline{n}|}=1. $$ The second identity follows by adding $v^n{}_np_x$ to both sides of the term insurance complement identity.

A Reliable Cross-Check Sequence

Before comparing numerical values, make sure both calculations describe the same contract. In particular, identify whether the benefit is paid at the end of the year of death or at another time, and whether the annuity is due or immediate. For a deferred contract, distinguish survival through the deferral period from survival during the subsequent coverage period.

Write the contract timeline.
Mark when each insurance benefit and annuity payment can occur, including whether a payment is conditional on being alive at the beginning or end of a year.

Calculate each value directly.
For insurance, use the probability of death in each covered year and discount to the actual payment time. For an annuity, use the probability of being alive at each payment time.

Apply the matching identity.
Use the same interest rate, survival assumptions, term, and payment timing in the identity as in the direct calculation.

Reconcile before rounding.
Compare the unrounded values. A discrepancy usually indicates a timing, survival, or contract-boundary mismatch rather than a rounding issue.

Worked Examples

Worked Example: Checking a Three-Year Term Insurance

Method: Calculate the death probabilities and annuity payments directly, then check their values using the term complement identity. A hypothetical unit term insurance has an effective annual interest rate of $i=25\%$. The conditional probabilities of surviving years 1, 2, and 3 are $0.80$, $0.75$, and $0.90$, respectively. Find $A^1_{x:\overline{3}|}$ and $\ddot{a}_{x:\overline{3}|}$, and verify the identity. Round final values to four decimal places.

Step 1: Calculate the interest quantities and cumulative survival probabilities. Here $v=1/1.25=0.80$ and $d=1-v=0.20$. Survival through years 1, 2, and 3 is $0.80$, $0.80(0.75)=0.60$, and $0.80(0.75)(0.90)=0.54$.

Step 2: Calculate the unconditional probability of death in each year. The conditional death probabilities are $0.20$, $0.25$, and $0.10$. Therefore the probabilities of death in years 1, 2, and 3 from the original age are $0.20$, $0.80(0.25)=0.20$, and $0.80(0.75)(0.10)=0.06$.

Step 3: Value the term insurance. Discount each benefit from the end of its year of death:

A^1_{x:\overline{3}|} =0.80(0.20)+0.80^2(0.20)+0.80^3(0.06) =0.16000+0.12800+0.03072 =0.31872.

Step 4: Value the temporary annuity-due. The three possible payments occur at times 0, 1, and 2. Their survival probabilities at those times are 1, $0.80$, and $0.60$:

\ddot{a}_{x:\overline{3}|} =1+0.80(0.80)+0.80^2(0.60) =1+0.64000+0.38400 =2.02400.

Step 5: Reconcile. The left side of the identity is $0.31872+0.20(2.02400)=0.31872+0.40480=0.72352$. The right side is $1-0.80^3(0.54)=1-0.27648=0.72352$. Both sides agree. Rounded to four decimal places, the term insurance is $0.3187$, the annuity-due is $2.0240$, and both sides of the identity are $0.7235$. This check avoids treating conditional annual death probabilities as unconditional probabilities.

Worked Example: Cross-Checking a Deferred Term Insurance

Method: Value the contract conditionally at the end of the deferral period, then multiply by the discounted probability of reaching that time. A hypothetical unit insurance defers coverage for two years and then provides two years of term coverage. The effective annual interest rate is $i=10\%$. The conditional annual survival probabilities for years 1 through 4 are $0.90$, $0.80$, $0.75$, and $0.50$. A temporary annuity-due is payable at times 2 and 3 if the life is alive at the respective payment time. Calculate both deferred values and check their relationship. Round final values to four decimal places.

Step 1: Set the deferral factor. Here $v=10/11$, $d=1/11$, and the probability of surviving the first two years is $0.90(0.80)=0.72$. The common discounted factor for reaching time 2 is $v^2(0.72)=(100/121)(0.72)=72/121=0.5950413$.

Step 2: Calculate values for the two-year coverage period, conditional on being alive at time 2. The conditional probabilities of death in coverage years 1 and 2 are $1-0.75=0.25$ and $0.75(1-0.50)=0.375$. Therefore the conditional term insurance value is

A^1_{x+2:\overline{2}|} =\frac{10}{11}(0.25)+\left(\frac{10}{11}\right)^2(0.375) =\frac{65}{121} =0.5371901.

The temporary annuity-due at time 2 has a payment immediately, followed by a payment one year later if the life survives that year:

\ddot{a}_{x+2:\overline{2}|} =1+\frac{10}{11}(0.75) =\frac{37}{22} =1.6818182.

Step 3: Convert the conditional values to deferred values. Each conditional value is multiplied by the same probability and discount factor for reaching time 2:

{}_{2|}A^1_{x:\overline{2}|} =\frac{72}{121}\frac{65}{121} =\frac{4680}{14641} =0.3196503, \qquad {}_{2|}\ddot{a}_{x:\overline{2}|} =\frac{72}{121}\frac{37}{22} =\frac{2664}{2662} =1.0007513.

Step 4: Check the conditional identity and scale it. Since $d=1/11$, $\frac{65}{121}+\frac{1}{11}\frac{37}{22} =\frac{130}{242}+\frac{37}{242}=\frac{167}{242}$. The conditional right side is $1-v^2(0.75)(0.50)=1-75/242=167/242$. Multiplying both sides by $72/121$ gives the deferred check:

{}_{2|}A^1_{x:\overline{2}|} +d\,{}_{2|}\ddot{a}_{x:\overline{2}|} =0.3196503+\frac{1}{11}(1.0007513) =0.4106277,

and $v^2(0.72)\{1-v^2(0.75)(0.50)\} =(72/121)(167/242)=0.4106277$. The values agree. Rounded to four decimal places, the deferred insurance value is $0.3197$, the deferred annuity value is $1.0008$, and both sides of the deferred identity are $0.4106$. The common mistake avoided here is applying the coverage-period survival assumptions as though they also described survival through the two-year deferral.

Worked Example: Checking a Three-Year Endowment Insurance

Method: Value the death benefits and survival benefit separately, and independently check the total using the annuity-due identity. A hypothetical three-year endowment insurance pays 1 at the end of the year of death or 1 at time 3 if the life survives. The effective annual interest rate is $i=5\%$, and the conditional annual survival probabilities are $0.90$, $0.80$, and $0.75$. Find the endowment value and verify it against the temporary annuity-due. Round final values to four decimal places.

Step 1: Calculate the interest and survival quantities. Here $v=20/21$ and $d=1/21$. The probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(0.25)=0.18$. The probability of survival through year 3 is $0.90(0.80)(0.75)=0.54$.

Step 2: Calculate the term insurance and pure endowment separately.

A^1_{x:\overline{3}|} =\frac{20}{21}(0.10) +\left(\frac{20}{21}\right)^2(0.18) +\left(\frac{20}{21}\right)^3(0.18) =0.4139942.

The survival benefit is $v^3{}_3p_x=(20/21)^3(0.54)=0.4664723$. Thus the endowment value, calculated from its benefit events, is $0.4139942+0.4664723=0.8804665$.

Step 3: Calculate the annuity-due and check the total. The annuity-due payments are at times 0, 1, and 2, with survival probabilities 1, $0.90$, and $0.90(0.80)=0.72$:

\ddot{a}_{x:\overline{3}|} =1+\frac{20}{21}(0.90) +\left(\frac{20}{21}\right)^2(0.72) =2.5102041.

The identity check gives $A_{x:\overline{3}|}+d\ddot{a}_{x:\overline{3}|} =0.8804665+(1/21)(2.5102041)=1.0000000$. Thus the event-by-event endowment value agrees with the value implied by the annuity-due. Rounded to four decimal places, the endowment value is $0.8805$, and the check sum is $1.0000$. The method avoids omitting the time-3 survival benefit or incorrectly including an annuity payment at time 3.

Common Mistakes and Their Corrections

A cross-check is useful only if the components have been defined consistently. The most frequent errors concern the distinction between a conditional probability for one year and a probability measured from the original age, or between a payment at the beginning and one at the end of a year.

Using a conditional death probability as an unconditional one. In the first example, the probability of death in year 2 is not $0.25$; it is $0.80(0.25)=0.20$, since the life must first survive year 1.
Discounting an annuity-due payment one year too far. Its first payment is at time 0, so its first present-value term is 1, not $v$. Later payments are discounted from their actual payment times.
Using the wrong survival probability for an annuity payment. A payment at time $k$ requires survival to time $k$; a payment at time $k-1$ uses survival only to time $k-1$.
Leaving out the endowment survival benefit. Term insurance alone covers death within the term. An endowment also pays if the life survives to the end, so its value includes $v^n{}_np_x$.
Applying an identity across mismatched periods. A deferred contract must account for survival and discounting during deferral before using the coverage-period identity.

For example, in the three-year term example, treating the conditional year-2 death probability $0.25$ as unconditional would give a year-2 insurance contribution of $0.80^2(0.25)=0.16000$, instead of the correct $0.80^2(0.20)=0.12800$. That incorrect contribution alone overstates the insurance value by $0.03200$. Using $0.75$ rather than the cumulative survival probability $0.60$ for the annuity payment at time 2 would also overstate the annuity. The identity will fail because the two mistaken calculations no longer describe consistent event probabilities.

Key takeaway: Cross-check values only after aligning interest, mortality, benefit timing, and coverage boundaries. For a term contract, compare its death-benefit value plus $d$ times the matching temporary annuity-due with $1-v^n{}_np_x$. For an endowment, include the pure endowment; the corresponding insurance value plus $d$ times the annuity-due must equal 1.

Check Your Understanding

For each problem, identify the payment times before calculating. Answers are rounded to four decimal places where appropriate.

A hypothetical one-year term insurance has $i=10\%$ and conditional survival probability $p_x=0.80$. Find $A^1_{x:\overline{1}|}$ and verify the complement identity using the one-payment annuity-due. (Answer: $A^1_{x:\overline{1}|}=0.1818$, $\ddot{a}_{x:\overline{1}|}=1.0000$, and both sides equal $0.2727$.)
A hypothetical two-year term insurance has $i=0\%$ and conditional survival probabilities $0.70$ and $0.50$. Find its insurance value and temporary annuity-due value, then check the identity. (Answer: $A^1_{x:\overline{2}|}=0.6500$, $\ddot{a}_{x:\overline{2}|}=1.7000$, and both sides equal $0.6500$.)
A hypothetical two-year endowment insurance has $i=20\%$ and conditional survival probabilities $0.80$ and $0.50$. Calculate its term insurance component, pure endowment component, and endowment value; then check with the annuity-due. (Answer: term component $0.4444$, pure endowment component $0.2778$, endowment value $0.7222$, annuity-due $1.6667$, and check sum $1.0000$.)
A hypothetical one-year deferred, one-year term insurance has $i=10\%$. The conditional survival probabilities are $0.90$ in year 1 and $0.50$ in year 2. A one-payment deferred annuity-due pays at time 1 if the life is alive then. Find the deferred insurance and annuity values and verify the scaled complement identity. (Answer: deferred insurance $0.3344$, deferred annuity-due $0.7438$, and both sides of the deferred identity equal $0.4013$.)

What You'll Learn