The General Discrete Retrospective Reserve Formula

Why the Retrospective Reserve Needs a Survival Adjustment

A retrospective reserve starts with cash flows that have already occurred. Premiums received build the policy fund, while benefits paid reduce it. To compare those past amounts at a common date, each cash flow must first be accumulated with interest. There is then one more adjustment: the accumulated balance for the original group of insureds must be expressed per policy still in force at the valuation date.

This tutorial develops that adjustment for a discrete annual model with premiums at the beginning of each policy year and death benefits at the end of the year of death. Premiums and death benefits may vary by policy year. The model excludes expenses and other cash flows; the reserve is calculated on the stated interest and mortality basis.

Definition: In the annual discrete model, a premium $P_k$ is paid at time $k$ if the insured is alive then. A death benefit $B_{k+1}$ is paid at time $k+1$ if death occurs during policy year $k+1$, the interval from time $k$ to time $k+1$. The retrospective reserve at duration $t$, denoted ${}_tV_x$, is the accumulated net amount from past premiums and benefits per policy in force at time $t$, calculated on the specified basis. It is measured immediately before any premium due at time $t$.

Write $i$ for the effective annual interest rate, $v=1/(1+i)$, and ${}_kp_x$ for the probability that a life aged $x$ survives $k$ years. Let ${}_1q_{x+k}$ be the probability of dying during the year after reaching age $x+k$. Thus, the probability of dying during policy year $k+1$, measured from issue, is ${}_kp_x\,{}_1q_{x+k}$. The probability of surviving to duration $t$ is

{}_tp_x=\prod_{k=0}^{t-1}{}_1p_{x+k}, \qquad {}_1p_{x+k}=1-{}_1q_{x+k}.

The General Discrete Formula

At duration $t$, a premium paid at time $k$ has earned interest for $t-k$ years. Only the fraction ${}_kp_x$ of the original group is expected to be alive to pay it. A benefit paid at time $k+1$ has earned interest for $t-k-1$ years after payment, and the probability of that payment is ${}_kp_x\,{}_1q_{x+k}$.

Theorem: For $t\geq 1$, under the annual discrete timing and cash-flow assumptions above, the retrospective reserve at duration $t$ is

{}_tV_x = \frac{ \displaystyle \sum_{k=0}^{t-1} P_k\,{}_kp_x(1+i)^{t-k} - \displaystyle \sum_{k=0}^{t-1} B_{k+1}\,{}_kp_x\,{}_1q_{x+k}(1+i)^{t-k-1} }{ {}_tp_x }.

The first sum is the expected accumulated value of premiums received from the original group. The second sum is the expected accumulated value of death benefits paid to the group. Their difference is divided by ${}_tp_x$, the expected fraction of the original group still insured at time $t$. The formula requires ${}_tp_x>0$; otherwise there are no expected in-force policies at the valuation date and the per-survivor reserve is undefined.

A premium at time $k$ is weighted by ${}_kp_x$, not by ${}_{k+1}p_x$: the insured needs to be alive at the beginning of that year to pay it. A death benefit for that year is weighted by ${}_kp_x\,{}_1q_{x+k}$, not by ${}_{k+1}p_x$: it is paid when the insured dies during the year.

Why the Formula Holds

The division by the survival probability follows from considering an expected cohort rather than treating the accumulated amount for one policy as the reserve. Imagine issuing $N$ identical policies. On the stated basis, the expected number alive at time $k$ is $N\,{}_kp_x$, and the expected number dying in policy year $k+1$ is $N\,{}_kp_x\,{}_1q_{x+k}$. Consequently, expected premiums paid at time $k$ are $N\,{}_kp_xP_k$, and expected benefits paid at time $k+1$ are $N\,{}_kp_x\,{}_1q_{x+k}B_{k+1}$.

Accumulate premiums.
Each expected premium amount paid at time $k$ grows to time $t$ by the factor $(1+i)^{t-k}$. Summing these amounts gives the first sum, multiplied by $N$.

Accumulate benefits.
Each expected death benefit paid at time $k+1$ grows to time $t$ by $(1+i)^{t-k-1}$. Summing these amounts gives the second sum, multiplied by $N$.

Allocate the balance to survivors.
The expected accumulated cohort balance is $N$ times the difference between the two sums. The expected number of policies in force at time $t$ is $N\,{}_tp_x$. Dividing the cohort balance by that survivor count cancels $N$ and gives the formula for ${}_tV_x$.

This is an expected-value cohort calculation under the specified interest and mortality assumptions. It is not a claim that the realized balance in an actual group will equal its expected value. Actual deaths and investment experience can differ from the basis used to calculate the reserve.

Worked Applications

Worked Example: Varying Premiums and Death Benefits

A hypothetical two-year policy has an annual effective interest rate of $5\%$. The mortality probabilities are ${}_1q_x=0.10$ and ${}_1q_{x+1}=0.20$. The premiums are $P_0=\$100$ at issue and $P_1=\$120$ at time 1 if the insured is alive. The death benefits are $B_1=\$500$ for death in year 1 and $B_2=\$600$ for death in year 2. Calculate ${}_2V_x$.

Method. Accumulate expected premiums and expected death benefits to time 2, then divide their difference by the probability of surviving two years.

Survival probabilities. The probability of reaching time 1 is ${}_1p_x=1-0.10=0.90$. The probability of reaching time 2 is ${}_2p_x=0.90(1-0.20)=0.90(0.80)=0.72$. The probability of dying in year 1 is $0.10$, and the probability of dying in year 2 is $0.90(0.20)=0.18$.

Accumulated premiums. The time-0 premium earns interest for two years: $\$100(1.05)^2=\$110.25$. The time-1 premium is paid only by the $0.90$ expected fraction alive then, and earns interest for one year: $0.90(\$120)(1.05)=\$113.40$. The total is $\$110.25+\$113.40=\$223.65$.

Accumulated benefits. Year-1 deaths receive $\$500$ at time 1, which accumulates one year: $0.10(\$500)(1.05)=\$52.50$. Year-2 deaths receive $\$600$ at time 2: $0.18(\$600)=\$108.00$. The total is $\$52.50+\$108.00=\$160.50$.

Therefore, ${}_2V_x=(\$223.65-\$160.50)/0.72=\$63.15/0.72=\$87.7083\ldots$, or $\$87.71$ to the nearest cent. The result is checked by multiplying back: $0.72(\$87.7083\ldots)=\$63.15$, the net accumulated cohort amount.

Common mistake avoided: The year-2 premium is weighted by $0.90$, the probability of being alive at time 1, rather than $0.72$, the probability of surviving all the way to time 2.

Worked Example: A Different Mortality Pattern

A hypothetical contract is valued at duration 2 using an effective annual interest rate of $4\%$. The probability of death in year 1 is $0.25$, and in year 2, conditional on reaching that year, it is $0.20$. The insurer receives $P_0=\$200$ at issue and $P_1=\$150$ at time 1 if the insured survives. The death benefits are $B_1=\$400$ and $B_2=\$500$. Find the retrospective reserve.

Method. Use the year-specific death probabilities to calculate the probabilities of each cash flow, then accumulate each to time 2.

The probability of surviving to time 1 is $0.75$, and the probability of surviving to time 2 is ${}_2p_x=0.75(0.80)=0.60$. The probabilities of death in years 1 and 2 are $0.25$ and $0.75(0.20)=0.15$, respectively.

The accumulated issue premium is $\$200(1.04)^2=\$200(1.0816)=\$216.32$. The expected time-1 premium is $0.75(\$150)=\$112.50$, which accumulates to $\$112.50(1.04)=\$117.00$. Total accumulated premiums are $\$216.32+\$117.00=\$333.32$.

The expected year-1 benefit is $0.25(\$400)=\$100$, accumulated one year to $\$104.00$. The expected year-2 benefit is $0.15(\$500)=\$75.00$, already at the valuation date. Total accumulated benefits are $\$104.00+\$75.00=\$179.00$.

Thus, ${}_2V_x=(\$333.32-\$179.00)/0.60=\$154.32/0.60=$$\$257.20$. Checking, $0.60(\$257.20)=\$154.32$, as required.

Common mistake avoided: The year-2 death probability from issue is $0.15$, not the conditional probability $0.20$. The latter must be multiplied by the probability $0.75$ of reaching year 2.

Worked Example: Matching Each Cash Flow to Its Date

A hypothetical three-year policy has an effective annual interest rate of $5\%$. Conditional death probabilities in years 1, 2, and 3 are $0.10$, $0.20$, and $0.25$. A premium of $\$100$ is paid at the start of each year if the insured is alive, and the death benefit is $\$300$ in every year. Calculate the reserve at time 3, before any premium due then.

Method. First find the probability of being alive at each premium date and the probability of death in each year. Then use the number of years each cash flow earns interest.

The survival probabilities are ${}_1p_x=0.90$, ${}_2p_x=0.90(0.80)=0.72$, and ${}_3p_x=0.72(0.75)=0.54$. The probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$.

The three accumulated expected premiums are $\$100(1.05)^3=\$115.7625$, $0.90(\$100)(1.05)^2=\$99.2250$, and $0.72(\$100)(1.05)=\$75.6000$. Their total is $\$290.5875$.

The three accumulated expected benefits are $0.10(\$300)(1.05)^2=\$33.0750$, $0.18(\$300)(1.05)=\$56.7000$, and $0.18(\$300)=\$54.0000$. Their total is $\$143.7750$.

The net accumulated cohort amount is $\$290.5875-\$143.7750=\$146.8125$. Therefore, ${}_3V_x=\$146.8125/0.54=$$\$271.875$, or $\$271.88$ to the nearest cent. The check $0.54(\$271.875)=\$146.8125$ recovers the net accumulated amount.

Common mistake avoided: The year-1 benefit is paid at time 1, so it earns interest for two years to time 3. It must not be accumulated for three years as though it were paid at issue.

Interpretation and Exam Traps

The formula keeps three distinct operations separate: determine whether a cash flow occurs, place it at its payment date, and accumulate it to the valuation date. Confusing any two of these can change the answer. A useful check is to verify that every premium probability refers to being alive when the premium is due, every death-benefit probability refers to dying in the specified year, and every interest exponent equals the time from payment to valuation.

For example, in the first worked example, a tempting but incorrect calculation is to weight the time-1 premium by the probability of surviving to time 2: $0.72(\$120)(1.05)$. This gives $\$90.72$, not the correct $\$113.40$, because the premium is paid at time 1 by anyone alive then, including those who may die during year 2. The correction is to use ${}_1p_x=0.90$. Conversely, the year-2 death benefit must use the probability of death in year 2 from issue, ${}_1p_x\,{}_1q_{x+1}$, not the probability of merely reaching time 1.

A reserve can be negative on a stated basis if accumulated expected benefits exceed accumulated expected premiums. That result signals a net outflow for the cohort under the assumptions; it does not justify changing the sign or replacing the denominator. At duration zero, both sums are empty and the retrospective reserve under this definition is zero.

Key takeaway: For a discrete retrospective reserve, accumulate each expected premium and benefit to the valuation date using its actual payment time. Subtract accumulated benefits from accumulated premiums, then divide by the probability of survival to the valuation date.

Check Your Understanding

For each problem, use the stated annual discrete timing and calculate the retrospective reserve at the stated duration.

A one-year policy has $i=5\%$, a year-1 death probability of $0.20$, a premium of $\$100$ at issue, and a death benefit of $\$300$ at time 1. Find ${}_1V_x$. (Answer: $\$56.25$)
A two-year policy has $i=0$, conditional death probabilities $0.10$ and $0.20$, premiums $P_0=\$50$ and $P_1=\$80$, and death benefits $B_1=\$100$ and $B_2=\$200$. Find ${}_2V_x$. (Answer: $\$105.56$ to the nearest cent)
A two-year policy has $i=10\%$, conditional death probabilities $0.20$ and $0.25$, premiums of $\$100$ at times 0 and 1 if alive, and death benefits of $\$200$ and $\$400$. Find ${}_2V_x$. (Answer: $\$141.67$ to the nearest cent)
A two-year policy has $i=0$, conditional death probabilities $0.10$ and $0.20$, premiums $P_0=\$100$ and $P_1=\$50$, and death benefits $B_1=\$300$ and $B_2=\$500$. Find ${}_2V_x$. (Answer: $\$34.72$ to the nearest cent)

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