Retrospective Reserves With Varying Premiums

Why Varying Premiums Require Survival Weights

A retrospective reserve measures the accumulated net cash flow associated with a policy cohort, divided by the number of policies expected to remain in force at the valuation date. When premiums depend on duration, each scheduled premium has its own amount. It also has its own probability of being received: a premium due several years after issue is paid only by policyholders who survive to that date.

The calculation therefore has two distinct tasks. First, accumulate each past premium and benefit to the valuation date. Second, weight each cash flow by the probability that it occurs in the original cohort, then divide the cohort’s expected accumulated net amount by the probability of surviving to the valuation date. The result is a reserve per policy in force at that date.

Definition: Consider an annual discrete contract issued at age $x$. Let $\Pi_k$ be the premium due at time $k$, paid at the beginning of policy year $k+1$ if the insured is then alive. Let the death benefit be $B$, payable at the end of the policy year of death. The retrospective reserve at time $t$ is measured immediately before any premium due at time $t$. Assume an effective annual interest rate $i$, no expenses or lapses, and a common interest and mortality basis for the calculation. Past premiums are those due at times $k=0,\ldots,t-1$.

The Cohort Formula With Duration-Dependent Premiums

The probability that a policy issued at age $x$ is in force at time $k$ is ${}_kp_x$. Thus, out of an original cohort of one policy on average, the expected premium received at time $k$ is ${}_kp_x\Pi_k$. That amount earns interest from time $k$ to time $t$, a period of $t-k$ years.

A death benefit paid at the end of policy year $k+1$ occurs with probability ${}_kp_xq_{x+k}$. It accumulates from time $k+1$ to time $t$, or $t-k-1$ years. Subtracting accumulated benefits from accumulated premiums gives the cohort’s expected accumulated net amount. Dividing by ${}_tp_x$ converts that amount to a value per policy surviving to time $t$.

{}_tV_x^{\mathrm{R}} = \frac{ \displaystyle\sum_{k=0}^{t-1} {}_kp_x\Pi_k(1+i)^{t-k} - \displaystyle\sum_{k=0}^{t-1} {}_kp_xq_{x+k}B(1+i)^{t-k-1} }{ {}_tp_x }.

The superscript $\mathrm{R}$ indicates a retrospective reserve. The formula is for the stated timing: premiums at the beginning of each policy year and death benefits at the end of the year of death. If a contract uses different payment dates or benefits, the accumulation periods and event probabilities must match those terms. In particular, a premium due at time $t$ is not included in this reserve, because the reserve is measured just before that payment.

Key interpretation: In the premium sum, ${}_kp_x$ weights $\Pi_k$ because that premium is collected only if the insured is alive at time $k$. In the death-benefit sum, ${}_kp_xq_{x+k}$ is the probability of surviving to the start of year $k+1$ and dying during that year. The denominator ${}_tp_x$ conditions the cohort result on being in force at the valuation time.

Why the Survival Weights Appear

The survival weights follow directly from taking expectations over the original cohort. For example, suppose there is one policy at issue. At time $k$, the premium $\Pi_k$ is received only on the event that the insured survives to time $k$. The expected premium cash flow at that time is therefore ${}_kp_x\Pi_k$, not simply $\Pi_k$. After accumulation to time $t$, its expected value is ${}_kp_x\Pi_k(1+i)^{t-k}$.

For a death benefit in year $k+1$, payment requires both survival to time $k$ and death during the following year. Under the annual life-contingency model, the probability of that event is ${}_kp_xq_{x+k}$. Accumulating the expected benefit payment to time $t$ gives ${}_kp_xq_{x+k}B(1+i)^{t-k-1}$. Summing these expected amounts over the past years gives the cohort’s expected accumulated net amount. The surviving fraction of the original cohort is ${}_tp_x$, so division by it yields the retrospective reserve per survivor.

Worked Example: Two Different Premiums Before the Valuation Date

A hypothetical policy pays a death benefit of $1,000 at the end of the year of death. The annual mortality probabilities are $q_x=0.10$ and $q_{x+1}=0.20$. Premiums of $200 at issue and $300 one year later are paid while the insured is alive. The effective annual interest rate is $i=5\%$. Calculate the retrospective reserve at time 2, immediately before any premium due then.

Method. Use the cohort formula. The survival probabilities needed are ${}_0p_x=1$, ${}_1p_x=1-0.10=0.90$, and ${}_2p_x=0.90(1-0.20)=0.72$.

Accumulated premiums. The time-0 premium accumulates for two years. The time-1 premium is collected only from the 90% surviving to time 1 and accumulates for one year:

$$ 200(1.05)^2 + 0.90(300)(1.05) = 220.50 + 283.50 = 504.00. $$

Accumulated benefits. Deaths in the first year have probability $0.10$; their benefits accumulate for one further year. Deaths in the second year have probability $0.90(0.20)=0.18$; their benefits are paid at time 2 and need no further accumulation:

$$ 0.10(1{,}000)(1.05) + 0.90(0.20)(1{,}000) = 105.00 + 180.00 = 285.00. $$

The expected accumulated net amount in the issue cohort is $504.00-285.00=219.00$. Divide by the fraction surviving to time 2:

{}_2V_x^{\mathrm{R}}=\frac{219.00}{0.72} =304.1666\ldots \approx \$304.17.

Check and trap avoided. The denominator uses the two-year survival probability, not the one-year probability. Also, the $300 premium is weighted by 0.90 because it is not collected from policyholders who die in the first year.

A Full Schedule With Three Varying Premiums

A useful calculation layout lists each premium and each benefit separately, along with its event probability and accumulation period. This prevents a change in premium amount from being confused with a change in the probability of payment. The following example uses a zero interest rate so that the survival weighting is visible without accumulation factors.

Worked Example: Three Premium Amounts and Constant Mortality

A hypothetical three-year contract has a $1,000 death benefit payable at the end of the year of death. The annual mortality probability is $q=0.10$ in each year, and the effective annual interest rate is $i=0\%$. Premiums due at times 0, 1, and 2 are $100, $200, and $400, respectively. Find the reserve at time 3, before any premium due at that time.

Method. First find the survival probabilities: ${}_1p_x=0.90$, ${}_2p_x=0.90^2=0.81$, and ${}_3p_x=0.90^3=0.729$.

The expected premiums received by the issue cohort are $100$ at time 0, $0.90(200)=180$ at time 1, and $0.81(400)=324$ at time 2. With zero interest, these amounts do not change when accumulated to time 3. Their total is $100+180+324=604$.

The probabilities of death in years 1, 2, and 3 are, respectively, $0.10$, $0.90(0.10)=0.09$, and $0.81(0.10)=0.081$. The expected benefits are therefore $100, $90, and $81, for a total of $100+90+81=271$.

The cohort’s expected accumulated net amount is $604-271=333$. Dividing by the 0.729 expected fraction surviving to time 3 gives

{}_3V_x^{\mathrm{R}}=\frac{333}{0.729} =456.7901\ldots \approx \$456.79.

Check and trap avoided. The three premiums are not simply added as $100+200+400$. The later amounts are paid by only a fraction of the original cohort. Weighting them gives expected premium receipts of $604, not $700.

Interest, Mortality, and Cash-Flow Timing Together

When both interest and mortality vary by duration, use the same procedure: determine the probability of each cash flow, identify its payment date, and accumulate it to the reserve date. The benefit payment period is one year shorter than the premium’s accumulation period when the premium is paid at the beginning of a year and the benefit at that year’s end. This timing difference is a frequent source of errors.

Worked Example: Varying Mortality, Interest, and Premiums

A hypothetical contract pays $1,500 at the end of the year of death. Annual mortality probabilities are $q_x=0.05$, $q_{x+1}=0.10$, and $q_{x+2}=0.20$. Premiums due at times 0, 1, and 2 are $250, $300, and $500. The effective annual interest rate is $i=4\%$. Calculate the retrospective reserve at time 3, before any premium due then.

Method. Calculate survival to each premium date and to the valuation date:

{}_1p_x=0.95,\qquad {}_2p_x=0.95(0.90)=0.855,\qquad {}_3p_x=0.855(0.80)=0.684.

The accumulated premiums are

$$ 250(1.04)^3 +0.95(300)(1.04)^2 +0.855(500)(1.04). $$

Evaluate the three terms: $250(1.124864)=281.216$; $0.95(300)(1.0816)=285(1.0816)=308.256$; and $0.855(500)(1.04)=427.5(1.04)=444.600$. The total accumulated premium amount is $281.216+308.256+444.600=1{,}034.072$.

The probabilities of death in the three years are $0.05$, $0.95(0.10)=0.095$, and $0.855(0.20)=0.171$. Accumulate each benefit from its payment date to time 3:

0.05(1{,}500)(1.04)^2 +0.095(1{,}500)(1.04) +0.171(1{,}500).

The terms are $75(1.0816)=81.120$, $142.5(1.04)=148.200$, and $256.5$. Total accumulated benefits are $81.120+148.200+256.500=485.820$. Thus the cohort’s expected accumulated net amount is $1{,}034.072-485.820=548.252$, and

{}_3V_x^{\mathrm{R}}=\frac{548.252}{0.684} =801.5380\ldots \approx \$801.54.

Check and trap avoided. The year-1 premium is weighted by 0.95, not by 0.90: it is due at time 1, so only survival through the first year determines whether it is paid. The year-1 death benefit, in contrast, is weighted by $0.95(0.10)$.

Common Errors and How to Avoid Them

The method is systematic, but a small timing or weighting error can affect several terms. Before calculating, write down the valuation instant and mark the payment date of every cash flow. Then check that the event probability and accumulation period agree with that date.

Ignoring survival for later premiums. A premium due at time $k$ is paid only if the insured survives to time $k$, so its cohort weight is ${}_kp_x$.
Using the wrong death probability. The probability of death during year $k+1$ is ${}_kp_xq_{x+k}$, not just $q_{x+k}$ when considering the original cohort.
Accumulating a benefit for too long. A benefit paid at the end of year $k+1$ accumulates for $t-k-1$ years to time $t$, while a premium paid at time $k$ accumulates for $t-k$ years.
Including the premium due at the valuation time. Under the convention here, the reserve is measured immediately before that premium. A reserve immediately after payment would require a different cash-flow definition.
Assuming the result must be positive. A retrospective reserve can be negative if accumulated premiums are insufficient to cover accumulated benefits on the specified basis. The arithmetic should still be checked against the stated contract and assumptions.

A concise check is to verify three items for every term: cash-flow amount, probability of payment, and years of accumulation. Finally, confirm that the denominator is the probability of being in force at the valuation time.

Check Your Understanding

For each problem, use the stated annual timing: premiums are paid at the beginning of a policy year while the insured is alive, benefits are paid at the end of the year of death, and the reserve is measured before any premium due at the valuation time.

A two-year hypothetical contract has a $500 death benefit, $q_x=q_{x+1}=0.20$, and $i=0\%$. Premiums are $50 at time 0 and $100 at time 1. Find the reserve at time 2. (Answer: $-\$78.13$)
A hypothetical contract has a $1,000 death benefit, $q_x=q_{x+1}=0.10$, and $i=10\%$. Premiums are $300 at time 0 and $400 at time 1. Find the reserve at time 2. (Answer: $\$690.12$)
A two-year contract pays $800 at the end of the year of death. Let $q_x=0.05$, $q_{x+1}=0.15$, and $i=0\%$; premiums are $100 at time 0 and $250 at time 1. Calculate the reserve at time 2. (Answer: $\$227.24$)
A three-year contract pays $1,000 at the end of the year of death. Let $q_x=0.10$, $q_{x+1}=0.20$, $q_{x+2}=0.25$, and $i=0\%$. Premiums are $100 at time 0, $150 at time 1, and $300 at time 2. Find the reserve at time 3. (Answer: $-\$16.67$)
A three-year contract has a $1,000 death benefit, $q=0.10$ in each year, and $i=0\%$. The premiums at times 0 and 1 are $100 and $200. What premium at time 2 produces a retrospective reserve of $300 at time 3? (Answer: $\$258.89$)

What You'll Learn