Diagnosing Sign and Timing Errors in Reserve Recursions

Why Reserve Recursions Are Easy to Misread

A one-year reserve recursion is a balance equation: funds available at the start of a policy year must be sufficient, in expectation, to meet the year’s benefit payments and the reserve needed for policies that remain in force. Many errors arise from changing the timing or conditioning in that balance without noticing. A premium received at the beginning of the year is not discounted like a year-end benefit, and a probability conditional on being alive at the start of the year is not generally an unconditional probability from issue.

The checks in this tutorial use the same basic model throughout. They are intended to help identify whether a proposed equation represents the stated cash flows, rather than to replace the prospective or retrospective reserve methods.

Definition: Consider a policy in force at time $k$, just before its annual premium is paid. Let ${}_kV_x$ be the reserve at that time, conditional on the insured being alive. A premium $P$ is paid immediately at time $k$. During the year, death has probability ${}_1q_{x+k}$, and survival to time $k+1$ has probability ${}_1p_{x+k}=1-{}_1q_{x+k}$. A death benefit $B$ is paid at the end of the year of death. If the insured survives, the reserve at time $k+1$, immediately before the next premium, is ${}_{k+1}V_x$. Assume an effective annual interest rate $i$, no expenses, and no other cash flows.

Under these assumptions, ${}_kV_x+P$ is available to earn interest for the year. The premium is collected at the start of the year; the death benefit and any reserve required for a surviving policy are measured at the end of the year. This timing convention is essential to the recursion below.

({}_kV_x+P)(1+i) = {}_1q_{x+k}B + {}_1p_{x+k}{}_{k+1}V_x

The equation is conditional on being alive at time $k$. Its right-hand side is the expected amount required at time $k+1$: the death benefit in the death outcome, or the next reserve in the survival outcome. Using $v=1/(1+i)$, the same equation can be written as

{}_kV_x = v\left({}_1q_{x+k}B+{}_1p_{x+k}{}_{k+1}V_x\right)-P.

A Short Derivation: Where the Premium Sign Comes From

The premium has a minus sign in the discounted form because it is money received by the insurer at time $k$, reducing the reserve that must already be held at that time. The sign is not a convention to memorize independently of the cash flows; it follows from the fund balance.

Set up funds at time $k$.
The insurer holds the reserve ${}_kV_x$ and receives the premium $P$ immediately. The total is ${}_kV_x+P$.

Accumulate to time $k+1$.
At effective annual interest rate $i$, those funds become $({}_kV_x+P)(1+i)$.

Average the end-of-year requirements.
Conditional on being alive at $k$, the expected requirement at $k+1$ is ${}_1q_{x+k}B+{}_1p_{x+k}{}_{k+1}V_x$. Equating the funds and requirements gives the recursion.

Isolate the reserve.
Divide by $1+i$, then subtract the premium received at time $k$. This gives ${}_kV_x=v({}_1q_{x+k}B+{}_1p_{x+k}{}_{k+1}V_x)-P$.

This derivation also indicates when the formula does not apply as written. If the premium is paid at the end of the year, if the benefit is paid immediately on death, or if expenses or other benefits occur, the cash-flow balance must reflect those different amounts and times. In particular, an end-of-year death benefit is discounted one full year in the discounted form; an immediate-on-death benefit cannot simply be inserted into this formula without an appropriate model for its payment time.

Worked Recursion Checks

Worked Example: Diagnosing a Premium-Sign Error

A hypothetical policy has a reserve of ${}_kV_x$ immediately before the start-of-year premium. The annual effective interest rate is $5\%$, the conditional probability of death during the year is $0.10$, and the death benefit of $\$1{,}000$ is paid at year-end. The conditional survival probability is $0.90$, and the reserve at the next anniversary before the next premium is $\$400$. The premium at time $k$ is $\$100$. Find the reserve at time $k$. State the method and check the balance.

Method. Apply the one-year fund-balance recursion. The end-of-year expected requirement is the death benefit weighted by the death probability plus the next reserve weighted by the survival probability:

$$ 0.10(1{,}000)+0.90(400)=100+360=460. $$

The funds available after receiving the premium must accumulate to $\$460$. Thus

({}_kV_x+100)(1.05)=460, $$ $$ {}_kV_x+100=\frac{460}{1.05}=438.095238\ldots, $$ $$ {}_kV_x=438.095238\ldots-100=338.095238\ldots.

Rounded to the nearest cent, the reserve is $\$338.10$. The check is $(338.095238\ldots+100)(1.05)=460$, matching the expected year-end requirement.

Common mistake avoided. Adding the premium after discounting gives $460/1.05+100=\$538.10$. That treats the premium as an amount the insurer must hold rather than as a receipt that helps fund the policy. The premium is received at time $k$, so it is included with the reserve before accumulation and subtracted when the equation is solved for the reserve.

Worked Example: Checking Premium and Benefit Timing

A hypothetical one-year policy has an annual effective interest rate of $25\%$. Conditional on being in force at the start of the year, death probability is $0.20$, survival probability is $0.80$, the year-end death benefit is $\$1{,}000$, and the next reserve for a survivor is $\$500$. A premium of $\$50$ is received at the start of the year. Calculate the current reserve and identify the error in discounting the premium.

Method. First calculate the expected end-of-year requirement:

$$ 0.20(1{,}000)+0.80(500)=200+400=600. $$

The current reserve plus the premium must accumulate to that amount:

({}_kV_x+50)(1.25)=600, $$ $$ {}_kV_x+50=\frac{600}{1.25}=480, $$ $$ {}_kV_x=480-50=430.

The reserve is $\$430.00$. To check, $(430+50)(1.25)=480(1.25)=600$, exactly the expected year-end requirement.

Common mistake avoided. Discounting the net amount $600-50$ gives $550/1.25=\$440$, which incorrectly treats the start-of-year premium as though it were paid at year-end. The correct timing is to add the premium to the reserve at time $k$, accumulate both for the year, and discount the end-of-year requirements before subtracting the time-$k$ premium. A premium due at a different time requires a balance equation for that different timing.

Worked Example: Recovering the Next Reserve Without Losing Survival Conditioning

At a policy anniversary, the current reserve immediately before premium is $\$200$. The premium of $\$40$ is paid immediately, the annual effective interest rate is $5\%$, and the conditional death probability is $0.20$. The death benefit of $\$500$ is paid at year-end. Find the next reserve for a survivor.

Method. Rearrange the standard recursion to isolate the survival-weighted next reserve:

{}_1p_{x+k}{}_{k+1}V_x = ({}_kV_x+P)(1+i)-{}_1q_{x+k}B.

Substitute the given values. The probability of survival is $1-0.20=0.80$:

({}_kV_x+P)(1+i)=(200+40)(1.05)=240(1.05)=252, $$ $$ {}_1q_{x+k}B=0.20(500)=100, $$ $$ 0.80{}_{k+1}V_x=252-100=152, $$ $$ {}_{k+1}V_x=\frac{152}{0.80}=190.

The next reserve is $\$190.00$. Substitution checks the result: $0.20(500)+0.80(190)=100+152=252$, equal to $(200+40)(1.05)$.

Common mistake avoided. Reporting $\$152$ as the next reserve stops at the survival-weighted amount. The recursion contains ${}_1p_{x+k}{}_{k+1}V_x$, not ${}_{k+1}V_x$ alone. To obtain the reserve conditional on survival, divide by the survival probability.

Worked Example: Converting Survival Probabilities to Conditional Mortality

For a hypothetical insured population, the probabilities of surviving from issue to durations two and three are ${}_2p_x=0.72$ and ${}_3p_x=0.60$. At duration two, the annual effective interest rate is $20\%$, the death benefit paid at the end of the year is $\$600$, the next reserve is $\$240$, and the start-of-year premium is $\$30$. Calculate the reserve at duration two.

Method. The recursion needs probabilities for the coming year conditional on survival to duration two. The conditional probability of surviving from duration two to duration three is

{}_1p_{x+2}=\frac{{}_3p_x}{{}_2p_x}=\frac{0.60}{0.72}=\frac{5}{6}.

Therefore the conditional death probability is ${}_1q_{x+2}=1-\frac{5}{6}=\frac{1}{6}$. The expected end-of-year requirement is

\frac{1}{6}(600)+\frac{5}{6}(240)=100+200=300.

Discount this amount one year and subtract the premium received immediately:

{}_2V_x=\frac{300}{1.20}-30=250-30=220.

The reserve at duration two is $\$220.00$. The balance check is $(220+30)(1.20)=300$, matching the conditional expected year-end requirement.

Common mistake avoided. The unconditional probability of surviving from issue to duration three, $0.60$, is not the probability of surviving the next year given survival to duration two. Similarly, the unconditional probability of dying between durations two and three is $0.72-0.60=0.12$, not the conditional one-year death probability. The conditional death probability is $(0.72-0.60)/0.72=1/6$. Using $0.12$ and $0.60$ directly in the one-year recursion would give a different, incorrectly conditioned balance.

A Practical Audit for Sign and Timing Errors

Before accepting a one-year recursion, write down when the reserve is measured, when the premium is paid, when the benefit is paid, and what is known about survival at the start of the year. Then check each term against those statements. This is particularly important when a problem gives survival probabilities from issue rather than one-year probabilities at the attained age.

Recursion feature	Diagnostic question	Standard model used here
Premium sign	Is the premium received by the insurer or paid out by it?	Received at time $k$; it adds to available funds.
Discounting	Is this cash flow at time $k$ or at time $k+1$?	End-of-year requirements are discounted one year; the start-of-year premium is not.
Benefit timing	When is the benefit paid in the stated model?	At the end of the year of death. Immediate payment requires different timing assumptions.
Survival conditioning	Are probabilities conditional on being alive at the start of this policy year?	Use ${}_1q_{x+k}$ and ${}_1p_{x+k}$; derive them from issue-based survival probabilities if needed.
Future reserve	Is the quantity a reserve conditional on survival or a probability-weighted amount?	${}_{k+1}V_x$ is conditional on survival; the recursion weights it by ${}_1p_{x+k}$.

A useful final check is to substitute the calculated reserve back into the original, undivided balance equation. The left side should equal the conditional expected amount required at the end of the year. This catches sign errors, missing discounts, and failure to divide by the survival probability when solving for a conditional next reserve.

Reserve recursions are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In exam calculations, a correct numerical answer is not enough if it comes from a cash-flow equation with inconsistent timing. Naming the reserve convention and checking the balance make the reasoning auditable.

Check Your Understanding

For each problem, use the stated reserve and cash-flow convention. Round money to the nearest cent where needed.

A hypothetical policy has $i=4\%$, conditional death probability $0.05$, survival probability $0.95$, a year-end death benefit of $\$2{,}000$, a next reserve of $\$800$, and a start-of-year premium of $\$100$. Find the current reserve. (Answer: $\$726.92$)
For a hypothetical policy, $i=5\%$, conditional death probability $0.10$, survival probability $0.90$, the year-end death benefit is $\$900$, the next reserve is $\$400$, and the start-of-year premium is $\$50$. Calculate the reserve and state why discounting the premium with the year-end requirements is incorrect. (Answer: $\$378.57$; the premium is received at the start of the year.)
A reserve of $\$250$ is measured immediately before a start-of-year premium of $\$25$. Let $i=4\%$, conditional death probability $0.20$, survival probability $0.80$, and the year-end death benefit be $\$400$. Find the next reserve conditional on survival. (Answer: $\$257.50$)
For a hypothetical insured, ${}_2p_x=0.70$ and ${}_3p_x=0.63$. At duration two, $i=10\%$, the year-end death benefit is $\$1{,}000$, the next reserve is $\$500$, and the start-of-year premium is $\$60$. Find the duration-two reserve using conditional one-year probabilities. (Answer: $\$440.00$)
At the final year of a hypothetical term policy, the reserve at the next anniversary after all benefits and premiums have been settled is zero. Let $i=6\%$, conditional death probability $0.04$, survival probability $0.96$, the year-end death benefit be $\$25{,}000$, and the start-of-year premium be $\$300$. Find the current reserve. (Answer: $\$643.40$)

Recursion feature	Diagnostic question	Standard model used here
Premium sign	Is the premium received by the insurer or paid out by it?	Received at time \(k\); it adds to available funds.
Discounting	Is this cash flow at time \(k\) or at time \(k+1\)?	End-of-year requirements are discounted one year; the start-of-year premium is not.
Benefit timing	When is the benefit paid in the stated model?	At the end of the year of death. Immediate payment requires different timing assumptions.
Survival conditioning	Are probabilities conditional on being alive at the start of this policy year?	Use \({}_1q_{x+k}\) and \({}_1p_{x+k}\); derive them from issue-based survival probabilities if needed.
Future reserve	Is the quantity a reserve conditional on survival or a probability-weighted amount?	\({}_{k+1}V_x\) is conditional on survival; the recursion weights it by \({}_1p_{x+k}\).

What You'll Learn