Diagnose Mixed-Timing Errors in Integrated Life-Contract Problems

Why Timing Audits Matter

A life-contract calculation can use correct mortality probabilities and a correct interest rate yet still produce a wrong answer if one cash flow is placed at the wrong time. A premium due at the beginning of a year is not discounted like a benefit paid at the end of that year. Similarly, a reserve conditional on being alive at a valuation date uses future mortality conditional on survival to that date, not probabilities measured from issue.

This tutorial develops a practical audit method for integrated life-contract calculations aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The examples use annual, discrete contracts. The effective annual interest rate is $i$, with $v=(1+i)^{-1}$. For each policy year $k$, $q_k$ is the probability of death during that year conditional on being alive at its beginning, and $p_k=1-q_k$.

Definition: A prospective reserve at a specified time is the expected present value, at that time and conditional on the insured being alive, of future benefits less future premiums. A reserve immediately before a premium includes that premium among future premiums; a reserve immediately after it does not. If a premium $P$ is due at that time, then $V^+=V^-+P$, or equivalently $V^-=V^+-P$, when the only change across the instant is receipt of that premium.

The signs in this convention describe the insurer's prospective liability, not a cash balance in a bank account. A negative before-premium reserve can occur when the premium due exceeds the value of the remaining benefits net of later premiums. In a net-premium calculation, the premium is determined by equating the expected present values of benefits and premiums at issue. The timing of each payment determines the factors in that equation.

A Reliable Audit Sequence

Before evaluating a long expression, translate the contract into a timeline. For each possible benefit, record the event that triggers it and the payment date. For each premium, record its due date and whether survival to that date is required. Then identify whether a reserve is measured just before or just after a premium. These checks prevent a common error: using a familiar annuity or insurance factor without checking whether its payment pattern matches the contract.

Mark the valuation time.
State whether the reserve is before or after any premium due at that instant, and condition on survival to the valuation time.

Place benefits on the timeline.
A benefit paid at the end of a year of death is discounted to that year-end. A maturity benefit is placed at the maturity date and paid only under the contract's maturity condition.

Place premiums on the timeline.
A premium at time zero is not discounted. A premium at time $k$ is included only if the insured is eligible to pay it at time $k$, and it is discounted $k$ years.

Reconcile independent calculations.
Compare the prospective value with a one-year reserve recursion when the timing convention permits it. A mismatch often identifies a boundary or survival-conditioning error.

Worked Examples: Locate the Payment Before Calculating

Worked Example: Two-Year Term Insurance and a Premium Due at Year One

A hypothetical two-year term policy pays $10,000 at the end of the year of death. Level net premiums $P$ are due at times 0 and 1 if the insured is alive. Assume $i=10\%$, $q_0=0.20$, and $q_1=0.25$, with each $q_k$ conditional on survival to the beginning of year $k$. Find $P$, then find the reserve at time 1 immediately before and after the premium due then.

Method. Value each benefit at its year-end and each premium at its due date. The probability of death in year 1 is $q_0=0.20$. The probability of death in year 2, measured from issue, is $p_0q_1=0.80(0.25)=0.20$. Thus

\operatorname{EPV}(\text{benefits}) =10{,}000(q_0v+p_0q_1v^2) =10{,}000\left(0.20\frac{10}{11}+0.20\frac{100}{121}\right) =10{,}000\left(\frac{22+20}{121}\right) =\$3{,}471.07.

The first premium is due at time 0. The second is due at time 1 only if the insured survives year 1, so its issue-date expected present value is $Pv p_0$. Therefore

\operatorname{EPV}(\text{premiums})=P(1+p_0v) =P\left(1+\frac{8}{11}\right) =P\frac{19}{11}.

Equating the two expected present values gives

P=\frac{420{,}000/121}{19/11} =\frac{420{,}000}{209} =\$2{,}009.57.

At time 1, conditional on being alive, only the year-2 death benefit remains. Immediately after the time-1 premium, there are no later premiums. Thus

{}_1V^+=10{,}000q_1v =10{,}000(0.25)\frac{10}{11} =\$2{,}272.73.

Immediately before that premium, it is still a future premium, so

{}_1V^-={}_1V^+-P =2{,}272.7273-2{,}009.5694 =\$263.16.

As a check, the issue-date premium value is $P(19/11)=\$3{,}471.07$, equal to the benefit value. The method avoids treating the time-1 premium as either due at time 0 or payable regardless of survival.

Worked Example: A Two-Year Endowment With a Maturity Payment

A hypothetical two-year endowment pays $12,000 at the end of the year of death if death occurs during the term, or $12,000 at time 2 if the insured survives to maturity. Level net premiums are due at times 0 and 1 while the insured is alive. Assume $i=5\%$, $q_0=0.10$, and $q_1=0.20$. Find the premium and the time-1 reserves before and after the premium.

Method. A year-1 death produces a payment at time 1 with probability $0.10$. At time 2, the contract pays in either of two mutually exclusive cases: death in year 2, or survival to maturity. Conditional on surviving year 1, those probabilities sum to $q_1+p_1=1$. From issue, the probability of a time-2 payment is therefore $p_0=0.90$. With $v=1/1.05=20/21$,

\operatorname{EPV}(\text{benefits}) =12{,}000(0.10v+0.90v^2) =12{,}000\left(\frac{2}{21}+\frac{40}{49}\right) =12{,}000\frac{134}{147} =\$10{,}938.78.

The premium at time 0 is certain. The time-1 premium is payable only if the insured survives year 1, so its expected present-value factor is $1+0.90v=13/7$. Hence

P=\frac{12{,}000(134/147)}{13/7} =\frac{1{,}608{,}000}{273} =\$5{,}890.11.

At time 1, if alive, the policy pays $12,000 at time 2 whether death occurs in year 2 or the insured survives to maturity. The after-premium reserve is therefore

{}_1V^+=12{,}000v =12{,}000\frac{20}{21} =\$11{,}428.57.

The before-premium reserve subtracts the premium due at time 1:

{}_1V^-={}_1V^+-P =11{,}428.5714-5{,}890.1099 =\$5{,}538.46.

A recursion provides an independent check. Immediately after the time-0 premium, the reserve is the value of benefits less the remaining time-1 premium, or $P$. One year later, the accumulated reserve must equal the year-1 death payment plus the survival-weighted reserve just before the next premium:

1.05P=0.10(12{,}000)+0.90({}_1V^-) =1{,}200+0.90(5{,}538.4615) =\$6{,}184.62.

Also, $1.05(5{,}890.1099)=\$6{,}184.62$, so the recursion agrees. The key protection against error is recognizing that the maturity payment is certain conditional on survival to time 1, even though the payment is triggered by either death or survival during the final year.

Worked Example: Conditional Reserves in a Three-Year Term Contract

A hypothetical three-year term policy pays $10,000 at the end of the year of death. Premiums $P$ are due at times 0, 1, and 2 if the insured is alive. Assume $i=10\%$, $q_0=0.10$, $q_1=0.20$, and $q_2=0.25$. Find $P$, then find the time-1 reserve immediately before and after the time-1 premium.

Method. From issue, the death probabilities in the three years are $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(0.25)=0.18$. Consequently

\operatorname{EPV}(\text{benefits}) =10{,}000(0.10v+0.18v^2+0.18v^3) =10{,}000\frac{499}{1331} =\$3{,}749.06.

Premiums at times 0, 1, and 2 have issue-date expected present-value factor $1+0.90v+0.72v^2$, since survival to time 2 has probability $0.90(0.80)=0.72$. Substituting $v=10/11$,

1+0.90v+0.72v^2 =1+\frac{9}{11}+\frac{72}{121} =\frac{292}{121}.

Equivalence gives

P=\frac{10{,}000(499/1331)}{292/121} =\frac{4{,}990{,}000}{3{,}212} =\$1{,}553.55.

Now value the contract at time 1 conditional on survival to time 1. From that point, the conditional probabilities of death in years 2 and 3 are $q_1=0.20$ and $p_1q_2=0.80(0.25)=0.20$. Thus the present value of future benefits is

10{,}000(0.20v+0.20v^2) =10{,}000\frac{42}{121} =\$3{,}471.07.

After the time-1 premium has been paid, only the time-2 premium remains. It is payable if the insured survives year 2, with conditional probability $p_1=0.80$, and is discounted one year from time 1. Therefore

{}_1V^+=3{,}471.0744-0.80vP =3{,}471.0744-\frac{8}{11}(1{,}553.5492) =\$2{,}341.09.

The before-premium reserve includes the time-1 premium as a future receipt:

{}_1V^-={}_1V^+-P =2{,}341.0890-1{,}553.5492 =\$787.54.

The distinction is both timing and conditioning: at time 1, the reserve calculation starts with the insured already alive, so it uses $q_1$, not the issue-to-time-2 death probability $p_0q_1$. The method also avoids counting the time-1 premium twice in the after-premium reserve.

Why the Prospective and Recursive Checks Agree

The one-year recursion is a useful audit because it follows the same cash flows as the prospective definition but groups them by what happens during the next year. Let $V_t^+$ be the reserve immediately after the premium at time $t$, and let $V_{t+1}^-$ be the reserve immediately before the next premium. Suppose a benefit $B_{t+1}$ is paid at the end of the year if death occurs during that year. Conditional on being alive at time $t$, the insurer's accumulated reserve at time $t+1$ must cover the death benefit in the death case or the next before-premium reserve in the survival case. Thus

(1+i)V_t^+=q_tB_{t+1}+p_tV_{t+1}^-.

This identity follows by splitting the possible outcomes over the year. Death occurs with probability $q_t$, producing the end-of-year benefit $B_{t+1}$. Survival occurs with probability $p_t$, leaving the insurer with the reserve $V_{t+1}^-$, which includes the premium due at that time. Both end-of-year amounts are compared with the accumulated reserve $(1+i)V_t^+$. If benefits differ by cause or timing, each must be represented separately; the displayed form applies to the stated single end-of-year benefit pattern.

A consistent audit applies the same convention on both sides. If the right side uses a reserve before the premium, do not substitute an after-premium reserve without adjusting by the premium. Likewise, if a formula is conditional on survival to time $t$, use probabilities conditional on that survival rather than probabilities from issue.

Common Mixed-Timing Errors

The following errors are especially easy to make in a multistep contract calculation:

Discounting a time-zero premium. A premium due at issue has factor 1, not $v$. A premium due at time 1 has factor $v$, with an additional survival probability if it is payable only while alive.
Using an unconditional death probability in a conditional reserve. At time $t$, the insured is already known to be alive. Use the mortality probabilities for future years conditional on being alive at their beginnings.
Confusing before-premium and after-premium reserves. Under the convention in this tutorial, $V^-=V^+-P$. Label the timing before substituting a reserve into a recursion.
Misplacing an endowment payment. A maturity benefit is paid at the maturity date if the insured survives to then. For an endowment that also pays on death, account for the distinct death and survival cases without double-counting.
Using a life annuity factor without checking eligibility. The premium factor must reflect when payments are due and whether the insured must survive to each due date. An immediate annuity and an annuity-due have different payment times.

Key takeaway: A correct life-contract chain is a timeline calculation. State the benefit date, premium date, survival condition, and reserve boundary at each step. Then check the result by a second identity or a one-year recursion, using exactly the same timing convention.

Check Your Understanding

For each problem, identify the payment timing and survival condition before calculating.

A hypothetical two-year term policy pays $5,000 at the end of the year of death. Premiums are due at times 0 and 1 while alive. If $i=0$, $q_0=0.10$, and $q_1=0.20$, find the level net premium. (Answer: $736.84)
A one-year policy pays $8,000 at the end of the year of death. At time 1, conditional on being alive, assume $q=0.30$ and $i=4\%$. There are no future premiums. Find the reserve immediately after the time-1 premium. (Answer: $2,307.69)
A hypothetical two-year endowment pays $6,000 at the end of the year of death, or at time 2 on survival to maturity. Premiums are due at times 0 and 1 while alive. If $i=0$, $q_0=0.20$, and $q_1=0.10$, find the net premium. (Answer: $3,333.33)
A three-year term policy pays $20,000 at the end of the year of death. At time 1, the insured is alive; $i=5\%$, $q_1=0.30$, and $q_2=0.20$. A $4,000 premium is due at time 2 if alive. Find the time-1 reserve immediately after the premium due at time 1 has been paid. (Answer: $5,587.30)
For the contract in Question 4, find the time-1 reserve immediately before the $4,000 premium due then, assuming the same premium amount is due at time 1. (Answer: $1,587.30)

What You'll Learn