Retrospective Reserves With Changing Benefits

Why Changing Benefits Matter

A policy may provide different death benefits in different policy years. For example, a hypothetical contract might pay a smaller amount during an initial period and a larger amount later. A retrospective reserve must reflect the benefit that applies in each year, rather than treating all historical claims as though they had the same amount.

The calculation has two distinct probability steps. First, determine the probability that a policy in the original cohort dies in each policy year; a later-year death requires survival through all earlier years. Then divide the cohort’s accumulated net cash-flow balance by the probability of surviving to the valuation date. This conditions the balance on the policy being in force at that date.

Definition: Consider an annual life insurance policy issued at age $x$. Let $B_j$ be the death benefit payable at the end of policy year $j+1$, if death occurs during that year, for $j=0,1,\ldots$. Let $P$ be a level annual premium, payable at the beginning of each policy year while the insured is alive. The retrospective reserve at time $k$ is the accumulated value of past premiums less accumulated death benefits for the original cohort, divided by the probability that a policy from that cohort survives to time $k$. We value the reserve just after the end-of-year-$k$ claim payments and before any premium due at time $k$.

Formula for Duration-Specific Death Benefits

Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Assume premiums are level, death benefits are paid at the end of the year of death, and the mortality probabilities for each policy year are $q_x,q_{x+1},\ldots$, with $p_{x+j}=1-q_{x+j}$. The benefit $B_j$ applies to a death during policy year $j+1$. There are no expenses, lapses, or other cash flows in this calculation.

A premium paid at time $j$ has $k-j$ years to accumulate to time $k$. A death benefit paid at time $j+1$ has $k-j-1$ years to accumulate. In the original cohort, the probability of being alive at time $j$ is ${}_jp_x$, so the probability of dying in policy year $j+1$ is ${}_jp_xq_{x+j}$. The probability of survival to time $k$ is ${}_kp_x$.

{}_kV_x= \frac{ P\displaystyle\sum_{j=0}^{k-1}{}_jp_x(1+i)^{k-j} - \displaystyle\sum_{j=0}^{k-1}{}_jp_xq_{x+j}B_j(1+i)^{k-j-1} }{{}_kp_x}

This formula assumes ${}_kp_x>0$, so there is a nonzero surviving cohort on which to condition. It also assumes the stated cash-flow timing. If premiums or benefits are paid at different times, their accumulation factors must be adjusted accordingly. The formula is not a license to use a single benefit amount when the contract specifies different amounts by duration.

Why the Cohort Formula Works

The formula follows by tracking the expected accumulated cash flows for the original group of policies. A premium contributes only for members alive when that premium is due. A claim contributes only for members who die in its applicable policy year. Dividing the resulting cohort balance by the fraction that survives gives the balance per surviving policy.

Accumulate premiums.
At time $j$, a fraction ${}_jp_x$ of the original cohort is alive to pay the premium $P$. Its time-$k$ accumulated contribution per original policy is $P{}_jp_x(1+i)^{k-j}$.

Accumulate claims.
The fraction dying in policy year $j+1$ is ${}_jp_xq_{x+j}$. Each such policy generates a benefit $B_j$ at time $j+1$, accumulated to time $k$ by $(1+i)^{k-j-1}$.

Condition on survival.
Subtract accumulated claims from accumulated premiums for the original cohort, then divide by ${}_kp_x$. This converts the cohort balance to a balance per policy surviving at time $k$.

For example, the first year’s claim contribution uses $q_xB_0$, while the second year’s uses ${}_1p_xq_{x+1}B_1=p_xq_{x+1}B_1$. The second-year death probability is conditional on reaching that year; the cohort probability must also include survival through the first year.

Worked Calculations

Worked Example: Two Years With an Increased Benefit

A hypothetical policy has $i=0.05$, level annual premium $P=\$1{,}500$, $q_x=0.20$, and $q_{x+1}=0.10$. The death benefit is $B_0=\$5{,}000$ in policy year 1 and $B_1=\$9{,}000$ in policy year 2. Find the retrospective reserve at time 2.

Method. Accumulate each premium and each year-specific claim to time 2 for the original cohort, then divide by the probability of surviving two years.

The annual survival probabilities are $p_x=1-0.20=0.80$ and $p_{x+1}=1-0.10=0.90$. Thus, ${}_2p_x=0.80(0.90)=0.72$.

The time-0 premium accumulates for two years: $1500(1.05)^2=1500(1.1025)=\$1{,}653.75$. The time-1 premium is paid by the fraction $0.80$ alive then, and accumulates for one year: $0.80(1500)(1.05)=\$1{,}260.00$. Total accumulated premiums are $\$1{,}653.75+\$1{,}260.00=\$2{,}913.75$.

The year-1 claim probability in the original cohort is $0.20$. Its accumulated claim amount is $0.20(5000)(1.05)=\$1{,}050.00$. The year-2 claim probability in the original cohort is $0.80(0.10)=0.08$. Its claim is paid at time 2, so it has no further accumulation: $0.08(9000)=\$720.00$. Total accumulated claims are $\$1{,}050.00+\$720.00=\$1{,}770.00$.

The cohort balance is $\$2{,}913.75-\$1{,}770.00=\$1{,}143.75$. Dividing by the surviving fraction gives \[ {}_2V_x=\frac{1143.75}{0.72}=\$1{,}588.5416\ldots \] Rounded to the nearest cent, the reserve is $\$1{,}588.54$.

Check. Multiplying the reserve by the surviving fraction gives $1588.5416\ldots(0.72)=\$1{,}143.75$, the accumulated cohort balance. The common mistake avoided here is using the second-year conditional death probability $0.10$ as though it were the cohort probability; the cohort probability is $0.80(0.10)=0.08$.

Worked Example: Three Distinct Benefit Amounts

A hypothetical policy has an effective annual interest rate $i=0.04$, a level annual premium $P=\$1{,}500$, and annual mortality probabilities $q_x=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. Its benefits are $B_0=\$4{,}000$, $B_1=\$7{,}000$, and $B_2=\$10{,}000$. Find the reserve at time 3.

Method. Use the applicable benefit and cohort death probability separately for each year. The survival probability to time 3 is ${}_3p_x=(0.90)(0.80)(0.75)=0.54$.

The three premium contributions, accumulated to time 3, are: at time 0, $1500(1.04)^3=1500(1.124864)=\$1{,}687.296$; at time 1, $0.90(1500)(1.04)^2=1350(1.0816)=\$1{,}460.16$; and at time 2, $(0.90)(0.80)(1500)(1.04)=1080(1.04)=\$1{,}123.20$. Their total is $\$4{,}270.656$.

The year-specific claim contributions are: $(0.10)(4000)(1.04)^2=\$432.64$; $(0.90)(0.20)(7000)(1.04)=0.18(7280)=\$1{,}310.40$; and $(0.90)(0.80)(0.25)(10000)=0.18(10000)=\$1{,}800.00$. The total is $\$3{,}543.04$.

The cohort balance is $\$4{,}270.656-\$3{,}543.04=\$727.616$. Therefore \[ {}_3V_x=\frac{727.616}{0.54}=\$1{,}347.437037\ldots \] Rounded to the nearest cent, the reserve is $\$1{,}347.44$.

Check. The first-year claim is accumulated for two years, the second-year claim for one year, and the third-year claim for zero years. The common mistake avoided is applying one benefit amount to all three claim probabilities, which would not represent this contract.

Worked Example: A Benefit That Decreases in the Final Year

A hypothetical three-year policy has $i=0.03$, level annual premium $P=\$900$, and mortality probabilities $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. The respective benefits are $B_0=\$6{,}000$, $B_1=\$8{,}000$, and $B_2=\$5{,}000$. Calculate the reserve at time 3.

Method. First calculate the fraction surviving to each premium date and the fraction dying in each year. Survival to time 3 is ${}_3p_x=(0.95)(0.90)(0.80)=0.684$. The cohort fractions dying in years 1, 2, and 3 are $0.05$, $(0.95)(0.10)=0.095$, and $(0.95)(0.90)(0.20)=0.171$.

Accumulated premiums are: $900(1.03)^3=900(1.092727)=\$983.4543$; $(0.95)(900)(1.03)^2=855(1.0609)=\$907.0695$; and $(0.95)(0.90)(900)(1.03)=769.5(1.03)=\$792.585$. Their sum is $\$2{,}683.1088$.

Accumulated claims are: $(0.05)(6000)(1.03)^2=300(1.0609)=\$318.27$; $(0.095)(8000)(1.03)=760(1.03)=\$782.80$; and $(0.171)(5000)=\$855.00$. Their sum is $\$1{,}956.07$.

The cohort balance is $\$2{,}683.1088-\$1{,}956.07=\$727.0388$. Hence \[ {}_3V_x=\frac{727.0388}{0.684}=\$1{,}062.9222\ldots \] The reserve, rounded to the nearest cent, is $\$1{,}062.92$.

Check. Multiplying $ \$1{,}062.9222\ldots$ by $0.684$ recovers the cohort balance of $\$727.0388$. The final-year benefit is $\$5{,}000$, not $\$8{,}000$; the benefit attached to the year of death must be read from the policy terms before calculating each claim contribution.

Common Exam Trap: Conditional Versus Cohort Probabilities

The mortality probability $q_{x+j}$ is conditional on being alive at the start of policy year $j+1$. The formula’s claim term uses the probability of that death within the original cohort, ${}_jp_xq_{x+j}$. Omitting ${}_jp_x$ overstates later-year claims because it counts deaths among people who would not have survived to that year.

In the first example, incorrectly using $0.10$ rather than $0.80(0.10)=0.08$ for the second-year cohort death probability gives accumulated claims of $0.20(5000)(1.05)+0.10(9000)=\$1{,}950$. The resulting incorrect reserve is $(2913.75-1950)/0.72=\$1{,}338.54$. The correct second-year claim contribution is $\$720$, not $\$900$, giving the correct reserve of $\$1{,}588.54$.

A second timing check is useful: a claim paid at the end of the valuation year is already at time $k$, so its accumulation factor is $1$. A claim paid earlier must earn interest from its payment date to time $k$. Premiums, by contrast, are paid at the beginning of their policy years and therefore accumulate for one more year than claims from the same policy year.

Key takeaway: For changing death benefits, match each policy year’s benefit $B_j$ with its cohort death probability ${}_jp_xq_{x+j}$, accumulate it from the end of that year to the valuation date, subtract all accumulated claims from accumulated premiums, and divide by ${}_kp_x$.

Check Your Understanding

For each problem, use the stated annual timing: premiums are paid at the beginning of each year while alive, and death benefits are paid at year-end. Round money to the nearest cent.

A two-year policy has $i=0.04$, $P=\$500$, $q_x=0.20$, and $B_0=\$2{,}000$. Find the retrospective reserve at time 1. (Answer: $\$150.00$)
A two-year policy has $i=0.05$, $P=\$600$, $q_x=0.10$, $q_{x+1}=0.20$, $B_0=\$3{,}000$, and $B_1=\$5{,}000$. Find the reserve at time 2. (Answer: $\$18.75$)
A two-year policy has $i=0.02$, $P=\$1{,}000$, $q_x=0.05$, $q_{x+1}=0.15$, $B_0=\$4{,}000$, and $B_1=\$7{,}000$. Find the reserve at time 2. (Answer: $\$1{,}000.50$)
A three-year policy has $i=0$, $P=\$1{,}500$, mortality probabilities $0.20, 0.10, 0.25$, and benefits $\$2{,}500,\$6{,}000,\$8{,}000$ in years 1, 2, and 3, respectively. Find the reserve at time 3. (Answer: $\$2{,}518.52$)
A two-year policy has $i=0.03$, $P=\$1{,}100$, $q_x=0.10$, $q_{x+1}=0.20$, $B_0=\$5{,}000$, and an unknown second-year benefit $B_1$. What value of $B_1$ produces a reserve of $\$400$ at time 2? (Answer: $\$7{,}687.17$)

What You'll Learn