Deferred Insurance Retrospective Reserves

Why Deferral Changes a Retrospective Reserve

A deferred life insurance policy has a period in which the contract is in force but a death benefit is not payable. Premiums may be collected during that period, and the policy may continue to collect them after coverage starts. A retrospective reserve must reflect both schedules: it accumulates premiums actually due and subtracts only benefits payable under the coverage at the time of death.

The key is to keep two questions separate: when is a premium paid? and when is a death covered? For a policy deferred for $n$ years, a death during the first $n$ years does not generate a claim. If coverage starts at time $n$, a death during policy year $n+1$ can generate a claim.

Definition: A deferred life insurance policy with an $n$-year deferment pays no death benefit if death occurs during the first $n$ policy years. Under the timing convention used here, coverage begins at time $n$: a death during year $j+1$ is covered only when $j \geq n$. A retrospective reserve at time $t$ is the accumulated value of the original cohort’s premiums minus its covered claims, divided by the number of policies from that cohort surviving to time $t$.

All examples use a discrete annual model. A premium due at time $j$ is paid by policies then in force; a death benefit for a death during year $j+1$ is paid at the end of that year. The reserve at time $t$ is measured after claims due at time $t$ have been settled and before a premium due at time $t$. There are no expenses, lapses, or changes in benefit amounts unless stated. Interest is credited at the stated effective annual rate $i$.

The Cohort Formula With a Deferment Period

Let ${}_jp_x$ be the probability that a policyholder aged $x$ survives $j$ years, and let $q_{x+j}$ be the probability of death during the following year, given survival to time $j$. Suppose a level benefit $B$ is payable at the end of the year of death, but only for deaths after the $n$-year deferment. Let $C_j$ denote the premium due at time $j$ per policy then in force; set $C_j=0$ in any year when no premium is due. For deferred whole-life coverage, the discrete retrospective reserve is

{}_tV_x^R = \frac{ \displaystyle\sum_{j=0}^{t-1} {}_jp_x C_j(1+i)^{t-j} - \displaystyle B\sum_{j=n}^{t-1} {}_jp_x q_{x+j}(1+i)^{t-j-1} }{ {}_tp_x }.

If $t \leq n$, the claim sum is empty: no covered death benefit has been paid by the valuation time. If coverage is temporary and ends after $r$ years of coverage, the claim sum includes only years $j=n,\ldots,n+r-1$, and only those years that have elapsed by time $t$. The premium sum should likewise include only premiums actually due before the valuation time.

The powers of $1+i$ reflect payment timing. A premium paid at time $j$ accumulates for $t-j$ years. A claim paid at the end of year $j+1$, at time $j+1$, accumulates for $t-j-1$ years. A claim paid exactly at time $t$ has no further accumulation.

Why the Formula Works

Consider an original cohort of $N$ identical policies. The expected number still in force at time $j$ is $N{}_jp_x$. Thus the expected premium collected at time $j$ is $N{}_jp_xC_j$. The expected number dying in year $j+1$ is $N{}_jp_xq_{x+j}$, so a covered death benefit produces an expected claim of $NB{}_jp_xq_{x+j}$ at time $j+1$. Accumulating those amounts to time $t$ gives the two sums in the numerator, each multiplied by $N$. The expected number surviving to time $t$ is $N{}_tp_x$. Dividing the accumulated cohort balance by this survivor count cancels $N$ and gives the formula.

This is a cohort expected-value reserve, not a claim that every realized group will have exactly the expected number of deaths. Its denominator is essential: the accumulated balance is shared across the policies from the original cohort that survive to the valuation date.

List the premiums.
Include premiums paid before time $t$, weighted by the probability the policy is in force when each premium is due.

List covered claims only.
Begin the claim sum with deaths in the first covered year, not deaths during the deferment.

Accumulate every cash flow to time $t$.
Use the premium’s payment time and the claim’s payment time to determine each accumulation period.

Divide by survivors.
Divide the cohort balance by ${}_tp_x$ to obtain the retrospective reserve per policy in force at time $t$.

Worked Calculations

Worked Example: Reserve at the End of the Deferment

A hypothetical policy defers a death benefit for two years. It charges a premium of $ \$64$ at times 0 and 1, provided the policyholder is alive. The benefit begins at time 2. Assume an effective annual interest rate of $i=10\%$, and a mortality model with $q_{x+j}=0.20$ for each year. Calculate the reserve at time 2. There have been no covered claims by that time.

Method. Accumulate the premiums paid by the original cohort to time 2, then divide by the fraction of that cohort surviving two years. Because time 2 is the end of the deferment, include no death claims.

The premium at time 0 accumulates for two years. The premium at time 1 is paid only by the $0.80$ expected fraction surviving to that time and accumulates for one year:

$$ 64(1.10)^2 + (0.80)(64)(1.10) = 77.44 + 56.32 = 133.76. $$

The fraction surviving to time 2 is ${}_2p_x=(0.80)^2=0.64$. Therefore,

{}_2V_x^R=\frac{133.76}{0.64}=209.00.

The reserve is $\$209.00$ per policy surviving to time 2. This check is consistent with the timing: both premiums have accumulated, but no death benefit has yet been payable.

Worked Example: Claims After Coverage Starts

A hypothetical deferred whole-life policy has a one-year deferment. It charges a level annual premium of $ \$128$ at times 0, 1, and 2 if the policyholder is alive. It pays $ \$1{,}000$ at the end of the year of death, but only for deaths after the deferment. Assume $i=25\%$ and $q_{x+j}=0.20$ in every year. Calculate the reserve at time 3.

Method. Accumulate premiums due at times 0, 1, and 2. Subtract claims for deaths in years 2 and 3 only: year 1 is within the deferment. Finally divide by the fraction surviving three years.

The accumulated premium contributions from the original cohort are

\begin{aligned} &128(1.25)^3 +(0.80)(128)(1.25)^2 +(0.80)^2(128)(1.25)\\ &=250.00+160.00+102.40 =512.40. \end{aligned}

Deaths during year 2 have probability $(0.80)(0.20)=0.16$ from the original cohort. Their benefit is paid at time 2, so it accumulates one year to time 3. Deaths during year 3 have probability $(0.80)^2(0.20)=0.128$; their benefits are paid at time 3 and do not accumulate further. The accumulated claims are

$$ (0.16)(1{,}000)(1.25)+(0.128)(1{,}000) =200+128 =328. $$

The accumulated cohort balance is $512.40-328=184.40$, and the survivor fraction is ${}_3p_x=(0.80)^3=0.512$. Thus

{}_3V_x^R=\frac{184.40}{0.512} =360.15625 \approx \$360.16.

The reserve is $\$360.16$ per surviving policy, rounded to the nearest cent. The claim sum starts with year 2, the first covered year, not year 1.

Worked Example: Premiums Stop When the Deferment Ends

A hypothetical policy has a two-year deferment and covers death from time 2 onward for a five-year term. Premiums of $ \$200$ are due only at times 0 and 1, if the policyholder is alive. A covered death benefit of $ \$5{,}000$ is paid at the end of the year of death. Assume $i=5\%$ and $q_{x+j}=0.10$ in each year. Calculate the reserve at time 3.

Method. Include the two premiums, but no later premiums because none are due. By time 3, only deaths during year 3 (the year with index $j=2$) have generated covered claims.

The accumulated premiums are

$$ 200(1.05)^3+(0.90)(200)(1.05)^2 =231.525+198.450 =429.975. $$

The fraction of the original cohort dying in year 3 is $(0.90)^2(0.10)=0.081$. The benefit is paid exactly at time 3, so it has no accumulation period. The cohort claim amount is therefore

$$ (0.081)(5{,}000)=405. $$

The survivor fraction is ${}_3p_x=(0.90)^3=0.729$. Subtract claims and divide by survivors:

{}_3V_x^R =\frac{429.975-405}{0.729} =\frac{24.975}{0.729} =34.259259\ldots \approx \$34.26.

The reserve is $\$34.26$ per surviving policy, rounded to the nearest cent. The five-year coverage period does not affect this calculation yet: at time 3, only its first covered year has elapsed.

Common Exam Trap: Treating the Deferment as Covered

A frequent error is to subtract death benefits for every death since policy issue, as though the policy were ordinary whole-life insurance from time 0. In the third example, that error would add claims for deaths in years 1 and 2. Accumulated to time 3, those incorrect claim amounts would be

(0.10)(5{,}000)(1.05)^2 +(0.09)(5{,}000)(1.05) =551.25+472.50 =1{,}023.75.

Adding the valid $ \$405$ claim gives an incorrect total of $ \$1{,}428.75$, and the resulting incorrect reserve would be $(429.975-1{,}428.75)/0.729\approx-\$1{,}370.06$. The arithmetic is consistent with that mistaken model, but the model is wrong: neither year 1 nor year 2 deaths is covered. The correct claim amount is $ \$405$, giving the reserve of $ \$34.26$.

A negative retrospective reserve can occur when the premiums accumulated to date are less than the claims accumulated to date. It should not, however, be accepted as a result without checking the benefit start date, payment timing, and survivor denominator. Those checks often reveal a timing error.

Key takeaway: Mark the deferment boundary on a timeline before writing the claim sum. Accumulate premiums from their due dates, accumulate claims from their payment dates, and divide the resulting cohort balance by the probability of surviving to the valuation time.

Check Your Understanding

For each problem, use the stated annual discrete timing convention and assume no expenses or lapses.

A policy defers coverage for three years. Premiums of $ \$50$ are paid at times 0 and 1 if alive. At time 2, assume $i=0\%$ and $q_{x+j}=0.10$ each year. No claims have been paid. Find the retrospective reserve. (Answer: $\$117.28$ per survivor.)
A one-year-deferred whole-life policy charges $ \$100$ at times 0 and 1 if alive and pays $ \$800$ for death after the deferment. At time 2, assume $i=0\%$ and $q_{x+j}=0.25$ each year. Find the reserve. (Answer: $\$44.44$ per survivor.)
A two-year-deferred policy charges $ \$100$ only at times 0 and 1. It pays $ \$2{,}000$ for death after deferment. At time 3, assume $i=10\%$ and $q_{x+j}=0.20$ each year. Find the reserve. (Answer: $-\$50.98$ per survivor.)
A one-year-deferred whole-life policy charges $ \$80$ at times 0, 1, and 2 if alive and pays $ \$1{,}000$ for death after deferment. At time 3, assume $i=5\%$ and $q_{x+j}=0.10$ each year. Find the reserve. (Answer: $\$88.52$ per survivor.)

What You'll Learn