Diagnose Boundary-Timing Errors in Premiums and Reserves

Why Boundary Timing Matters

A reserve calculation can use the correct mortality rates, interest rate, benefit amounts, and premiums and still be wrong because one cash flow has been placed at the wrong time. The most common boundary errors occur at a policy anniversary: a premium due then may be included in a reserve measured just after payment, a death benefit from the preceding year may be carried into the reserve for a survivor, or a benefit may be discounted one period too many.

The remedy is to identify the valuation instant precisely. In this tutorial, annual premiums are due at integer policy anniversaries while the insured is alive. A death benefit for death during a policy year is paid at the end of that year. We distinguish the reserve immediately before an anniversary premium from the reserve immediately after it.

Definition: At duration $k$, let $V_k^-$ be the prospective reserve immediately before a premium due at time $k$, and let $V_k^+$ be the prospective reserve immediately after that premium is paid. If the premium due at time $k$ is $P_k$, the insured is alive, and there are no other cash flows at that instant, then $V_k^- = V_k^+ - P_k$. The notation ${}_kV_x$ is often used for a reserve at duration $k$; always state whether it is measured before or after a premium.

This difference is a matter of cash-flow timing, not a change in the policy’s benefits or mortality. Before payment, the premium is still an incoming future cash flow from the insurer’s perspective, so it reduces the prospective reserve. After payment, that premium is no longer future. The relationship assumes that the premium is actually due at time $k$; if premiums have ended, or the contract is not in force, there is no such premium to subtract.

Set the Timeline Before Calculating

For annual calculations, let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $q_k$ be the probability that a life alive at duration $k$ dies during policy year $(k,k+1]$, and let $p_k=1-q_k$. Suppose a death benefit $B_{k+1}$ is paid at time $k+1$ for a death in that year. If a life survives to time $k+1$, its reserve at that instant is first considered immediately before any premium due there.

V_k^+ = v\left(q_k B_{k+1}+p_k V_{k+1}^-\right)

This recursion is for a discrete annual model with end-of-year death benefits and no other payment at the end of the year. The death branch contributes the benefit at time $k+1$; the survival branch contributes the reserve just before the next premium. If the contract also pays a maturity benefit on survival, that payment must be included in the survival branch. If a death benefit is paid immediately at death rather than at the end of the year, its discount timing must be changed.

Name the valuation instant.
Record the policy duration and whether the reserve is before or after any premium due at that time.

Separate death and survival.
For the coming policy year, identify the benefit on death and the reserve reached on survival.

Discount each branch once.
In the annual recursion, both outcomes are valued at time $k+1$, then discounted to time $k$.

Handle the anniversary premium separately.
Convert between $V_{k+1}^-$ and $V_{k+1}^+$ only after checking that a premium is due and specifying which side of it the reserve uses.

Why the Recursion Has These Terms

The recursion follows by conditioning on what happens during the next policy year. At time $k$, immediately after any premium due then, there are two mutually exclusive outcomes. With probability $q_k$, death occurs during the year and the insurer pays $B_{k+1}$ at time $k+1$. With probability $p_k$, the life survives and the insurer reaches time $k+1$ with a reserve measured before the next premium. Thus the expected value at time $k+1$ is $q_kB_{k+1}+p_kV_{k+1}^-$. Discounting one year gives the displayed formula.

The superscript on the next reserve matters. If a premium $P_{k+1}$ is due at time $k+1$, substituting $V_{k+1}^+$ without adjustment would omit that premium from the future cash flows. Since $V_{k+1}^-=V_{k+1}^+-P_{k+1}$, an equivalent recursion is

V_k^+ = v\left(q_k B_{k+1}+p_k\left(V_{k+1}^+-P_{k+1}\right)\right).

This equivalent form applies only when the premium $P_{k+1}$ is due at time $k+1$ on survival. It is not correct to subtract a premium that is not due, for example after the premium-paying term has ended.

Worked Example: Is the Anniversary Premium Still Future?

A hypothetical two-year term policy pays $1,000 at the end of the year of death. The effective annual interest rate is 5%. The conditional death probabilities are $q_0=0.10$ and $q_1=0.20$. A level premium $P$ is due at times 0 and 1 if the insured is alive. Find $P$, then find the reserve at time 1 immediately before and immediately after the second premium.

Method. Use the equivalence principle at issue, then distinguish the two reserve instants at time 1. The common error is to include the time-1 premium in the reserve after it has already been paid.

The discount factor is $v=1/1.05=20/21$. The expected present value of benefits is

1000\left(q_0v+p_0q_1v^2\right) =1000\left(0.10\cdot\frac{20}{21}+0.90\cdot0.20\cdot\left(\frac{20}{21}\right)^2\right) =1000\left(\frac{2}{21}+\frac{8}{49}\right) =\frac{38{,}000}{147} =\$258.5034\ldots

The expected present value of premiums per dollar of premium is $1+p_0v$: one premium is due at time 0, and the time-1 premium is due only if the insured survives the first year. Therefore,

P=\frac{38{,}000/147}{1+0.90(20/21)} =\frac{38{,}000/147}{13/7} =\frac{38{,}000}{273} =\$139.19

At time 1, immediately after payment, only the second year of coverage remains. The probability of death during that year is $0.20$, and the benefit is paid one year later. Thus

V_1^+=v(0.20)(1000)=\frac{4{,}000}{21}=\$190.48.

Immediately before the time-1 premium, that premium is still a future incoming cash flow, so

V_1^-=V_1^+-P =\frac{4{,}000}{21}-\frac{38{,}000}{273} =\frac{14{,}000}{273} =\$51.28.

As a check, the issue reserve immediately before the time-0 premium is zero: the premium was set by equivalence. Immediately after that premium, the reserve is $139.19, equal to the present value of future net costs. All dollar amounts are rounded to the nearest cent; intermediate values are unrounded.

Common mistake and correction. Reporting $V_1^+=\$51.28$ incorrectly carries the time-1 premium into a reserve measured after payment. The correction is to add the premium back: $V_1^+=V_1^-+P=\$51.28+\$139.19=\$190.47$, or $\$190.48$ using unrounded values.

Do Not Drop the Death Branch

When working backward from a later reserve, the death benefit for the year being valued is a separate cash flow. A survivor’s reserve does not represent the claim paid to someone who dies during that year. Omitting the death branch understates the reserve by the discounted expected death benefit for that interval.

Worked Example: Recover a Reserve Without Losing the Benefit

A hypothetical two-year policy pays $1,200 for death during the first year and $2,400 for death during the second year. The effective annual interest rate is 4%. The conditional death probabilities are $q_0=0.10$ and $q_1=0.25$. A premium of $80 is due at time 1 if the insured is alive; no later premium is due. Find the reserve at time 0 immediately after the initial premium has been paid.

Method. First calculate the time-1 reserve on each side of the premium, then apply the recursion from time 0. The common error is to use only the survival branch and omit the first-year death benefit.

At time 1 immediately after the premium, the remaining benefit is $2,400, payable at time 2 if death occurs during the second year. Hence

V_1^+=\frac{1}{1.04}(0.25)(2400)=\$576.92.

The time-1 premium is due immediately before this after-premium reserve, so the before-premium reserve is

V_1^-=V_1^+-80=\$576.9231\ldots-\$80=\$496.92.

At time 0, a death in the first year produces a $1,200 payment at time 1. Survival leads to $V_1^-$. Therefore,

V_0^+ =\frac{1}{1.04}\left((0.10)(1200)+(0.90)(496.9231\ldots)\right) =\frac{1}{1.04}(120+447.2308\ldots) =\$545.41.

For a check on the death branch, its contribution to $V_0^+$ is $v(0.10)(1200)=\$115.38$. If that branch is omitted, the calculation gives $v(0.90)(496.9231\ldots)=\$430.03$, which is too low by exactly $115.38. The reported reserves are rounded to the nearest cent.

Common mistake and correction. The incorrect survival-only calculation treats the reserve on survival as if it covered deaths during the year. It does not: the death benefit is paid on a different, mutually exclusive branch. Include $q_0B_1$ and $p_0V_1^-$ before discounting.

Place Each Anniversary Cash Flow on Its Correct Side

At an anniversary, the reserve for a surviving policyholder is not a record of every payment made during the preceding year. In the annual model used here, deaths during $(k-1,k]$ receive the benefit at time $k$. A surviving policyholder reaches time $k$ without receiving that death benefit and may then owe the next premium. The reserve for that survivor is calculated from the remaining coverage and future cash flows.

A simple timeline helps prevent both errors: write the death benefit at the end of the year in which death is covered; mark the anniversary premium at the instant it is due; then label the reserve immediately before or after that premium. A benefit paid at time $k$ for a death in the preceding year is not discounted from time $k+1$, and it is not added to the future reserve of a life known to have survived to time $k$.

Worked Example: Different Benefits on Opposite Sides of an Anniversary

A hypothetical two-year policy pays $1,000 at the end of the first policy year if death occurs in that year, and $2,000 at the end of the second year if death occurs in that year. The effective annual interest rate is 5%; $q_0=0.10$ and $q_1=0.20$. Premiums of $100 at time 0 and $150 at time 1 are due while the insured is alive. Calculate $V_1^+$, $V_1^-$, and $V_0^+$.

Method. Start with the surviving life at time 1, after the first-year claim period has ended, and then work backward. The common error is to carry the first-year benefit into the reserve for a survivor or to place its payment at time 2.

After the time-1 premium has been paid, only the second-year benefit remains. Its expected value at time 1 is

V_1^+=\frac{1}{1.05}(0.20)(2000) =\frac{8000}{21} =\$380.95.

Before the time-1 premium, the premium is still due, giving

V_1^-=V_1^+-150 =\$380.9524\ldots-\$150 =\$230.95.

At time 0 immediately after the initial premium, death in the first year leads to a $1,000 payment at time 1. Survival leads to $V_1^-$. Thus

V_0^+ =\frac{1}{1.05}\left((0.10)(1000)+(0.90)(230.9524\ldots)\right) =\frac{1}{1.05}(100+207.8571\ldots) =\$293.20.

The first-year benefit’s contribution to $V_0^+$ is $v(0.10)(1000)=\$95.24$, because it is payable at time 1. If it is mistakenly assigned to time 2, its contribution becomes $v^2(0.10)(1000)=\$90.70$, an understatement of $4.54. The reserve at time 1 is for a life that has survived to time 1, so it includes neither that first-year claim nor a second discount of its payment. Amounts are rounded to the nearest cent.

Common mistake and correction. Do not move a cash flow across the anniversary merely because the calculation is being done at that anniversary. The first-year death benefit is on the death branch ending at time 1; the second-year benefit belongs to the next interval. Keep the branches and payment dates separate.

Boundary Checks to Use in Practice

Before accepting a premium or reserve result, check the following points:

Premium due now: If the valuation is immediately before a premium, include that premium as a future incoming cash flow. If the valuation is immediately after payment, do not include it again.
Death benefit in the interval: The death branch must include the benefit covered by the policy year being valued, discounted according to its payment date.
Survivor at the anniversary: Use the reserve appropriate to a life that reaches the anniversary. Do not add a benefit already paid on the alternative death branch.
Premium-paying term: Subtract an anniversary premium only if it is due then and the policy is in force.
One discount per year: A cash flow at time $k+1$ is discounted once to time $k$. A cash flow at time $k$ is not discounted an additional year.

These checks are especially useful when premium equivalence, a prospective reserve, and a recurrence appear in one calculation. The equations can be algebraically consistent while still describing different valuation instants, so label the timing before combining them. This approach is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Key takeaway: At every policy anniversary, state whether the reserve is before or after the premium. In the annual recursion, include the benefit on the death branch and the next reserve on the survival branch, using the payment date and reserve timing that actually apply.

Check Your Understanding

For each problem, identify the valuation instant before calculating. Unless stated otherwise, death benefits are paid at the end of the policy year of death.

A one-year term policy has $i=5\%$, $q_0=0.10$, a $1,000 death benefit, and a $50 premium due at time 0. Find $V_0^+$ and $V_0^-$. (Answer: $V_0^+=\$95.24$, $V_0^-=\$45.24$.)
At duration 1, a policy has one year of coverage remaining, $i=5\%$, $q_1=0.20$, and a $1,500 benefit. A $60 premium is due at time 1. Find the reserve immediately before and after that premium. (Answer: $V_1^+=\$285.71$, $V_1^-=\$225.71$.)
For the coming year, $i=4\%$, $q_k=0.10$, the death benefit is $800, and the reserve at the next anniversary immediately before premium is $120. Find $V_k^+$. (Answer: $V_k^+=\$180.77$.)
A two-year term policy pays $1,000 at the end of the year of death. The effective annual rate is 25%, with $q_0=0.20$ and $q_1=0.10$. Level premiums are due at times 0 and 1 while alive. Find the level premium by equivalence. (Answer: $P=\$128.78$.)
A surviving policyholder at time 1 has one year of coverage remaining. The effective annual rate is 4%, $q_1=0.25$, the death benefit is $1,600, and a $100 premium is due at time 1. Find the reserves immediately after and immediately before that premium. (Answer: $V_1^+=\$384.62$, $V_1^-=\$284.62$.)

What You'll Learn