Recursions for Duration-Varying Premiums

Why Premium Duration Matters

A reserve recursion balances the premium collected at a particular policy duration against expected benefits and the reserve needed at the end of the year. When annual premiums change over time, the premium at duration $k$ is not interchangeable with the premium at any other duration. Using the wrong amount can produce an incorrect reserve even when the mortality and interest calculations are otherwise correct.

This tutorial considers annual net premium reserves for a life insurance contract. The key calculation is local to each policy year: use the premium scheduled for that year, then move to the reserve at the next duration. A contract with the same premium at every paying duration is the special level-premium case; it should not be assumed when the schedule specifies different amounts.

Definition: Let $P_k$ be the annual premium due at duration $k$, conditional on the insured being alive at that time. A duration-varying premium schedule is a sequence $P_0,P_1,\ldots$ whose amounts may differ by duration. A level-premium schedule has $P_k=P$ at every premium-paying duration.

Reserve Timing and the Annual Model

We use a term insurance model with annual policy years. At time $k$, the insured is alive and the premium $P_k$ is due immediately. A death during policy year $k$ produces benefit $B_{k+1}$ payable at the end of that year. If the insured survives to time $k+1$, the contract continues and the next premium is due then. Expenses are excluded, so these are net reserves.

Write ${}_kV_x^-$ for the reserve at duration $k$, immediately before collecting the premium due at that time. Write ${}_kV_x^+$ for the reserve immediately after that premium is collected. With this definition, ${}_kV_x^+={}_kV_x^-+P_k$. This relation is a timing identity: the post-premium reserve includes the current premium, whereas the pre-premium reserve does not.

Let $i$ be the effective annual interest rate, and let ${}_1q_{x+k}$ and ${}_1p_{x+k}$ be the probabilities of death and survival, respectively, during the year from age $x+k$. Thus ${}_1p_{x+k}=1-{}_1q_{x+k}$. The recursion below uses the pre-premium reserve at both endpoints of a policy year.

One-year reserve recursion: Under annual premium payments at the beginning of each policy year, an end-of-year death benefit $B_{k+1}$, and no expenses, the pre-premium reserves satisfy $$ \left({}_kV_x^-+P_k\right)(1+i) ={}_1q_{x+k}B_{k+1}+{}_1p_{x+k}\,{}_{k+1}V_x^-. $$ Equivalently, $$ {}_kV_x^-= \frac{{}_1q_{x+k}B_{k+1}+{}_1p_{x+k}\,{}_{k+1}V_x^-}{1+i}-P_k. $$

For an $n$-year term contract with no maturity benefit and no premium due after expiry, the terminal condition is ${}_nV_x^-=0$. If a contract has a maturity benefit, the endpoint cash flow must reflect it; the zero terminal condition would not apply unchanged. Likewise, the displayed recursion assumes the premium is due at the beginning of the year and the death benefit is paid at year-end. Different payment timing requires a different equation.

Why the Recursion Holds

At duration $k$, the reserve plus the premium just collected is ${}_kV_x^-+P_k$. Accumulated for one year, this amount is $({}_kV_x^-+P_k)(1+i)$. At year-end, death occurs with probability ${}_1q_{x+k}$, in which case the insurer pays $B_{k+1}$. Survival occurs with probability ${}_1p_{x+k}$, in which case the continuing contract requires the reserve ${}_{k+1}V_x^-$, measured just before the next premium is collected. These two outcomes give the expected year-end amount ${}_1q_{x+k}B_{k+1}+{}_1p_{x+k}{}_{k+1}V_x^-$.

Equating the accumulated amount to the expected amount required at year-end gives the recursion. Rearranging and discounting by $1+i$ gives the reserve form. This argument also explains why $P_k$ appears: it is the premium due now, at duration $k$. The next premium is already accounted for by defining the next reserve immediately before it is collected.

Calculating Reserves Backward With a Changing Schedule

For a finite contract with a known terminal reserve, the usual method is to start at expiry and apply the recursion one year at a time in reverse order. At each step, use the premium $P_k$ for that specific duration. Do not replace the schedule with an average premium or carry the first-year premium into later years unless the contract actually specifies a level premium.

Worked Example: A Two-Year Contract With Unequal Premiums

A hypothetical two-year term policy pays $B=1050$ at the end of the year of death. The effective annual interest rate is $i=0.05$, and the one-year death probability is $0.10$ in each policy year. Premiums are $P_0=80$ and $P_1=50$, payable at the beginning of the corresponding year. There is no maturity benefit, so ${}_2V_x^-=0$. Find the reserves immediately before the premiums at durations 1 and 0. Give the final reserves to the nearest cent.

Method. Apply the recursion backward from duration 2. The common mistake is to use the first-year premium of 80 again at duration 1, despite the stated premium of 50.

At duration 1, the death probability is $0.10$, the survival probability is $0.90$, and the next reserve is zero:

({}_1V_x^-+50)(1.05) =0.10(1050)+0.90(0)=105.

Therefore ${}_1V_x^-+50=105/1.05=100$, so ${}_1V_x^-=50$.

At duration 0, use $P_0=80$ and the reserve just calculated:

({}_0V_x^-+80)(1.05) =0.10(1050)+0.90(50) =105+45=150.

Thus ${}_0V_x^-+80=150/1.05=142.857142\ldots$, giving ${}_0V_x^-=62.857142\ldots$. Rounded only at the final step to two decimals, the reserves are ${}_1V_x^-=\$50.00$ and ${}_0V_x^-=\$62.86$.

Check. At duration 0, the post-premium amount $62.857142\ldots+80$, accumulated at 5%, is $150$, equal to the expected year-end amount $105+0.90(50)=150$. If the wrong premium of 80 were used at duration 1, the calculated reserve there would be $100-80=20$, not 50.

A Three-Year Backward Calculation

The same procedure applies when mortality, benefits, or premiums vary by duration. Each year’s equation uses the probabilities and benefit for that year, as well as its own premium amount. The terminal condition starts the calculation; it does not imply that the premiums in earlier years are equal.

Worked Example: Tracking Three Premium Amounts

A hypothetical three-year term policy pays $B=1000$ at the end of the year of death. The effective annual interest rate is $i=0.05$, and the one-year death probability is $0.10$ in each year. The premiums are $P_0=30$, $P_1=40$, and $P_2=50$. There is no maturity benefit, and ${}_3V_x^-=0$. Calculate the pre-premium reserves at durations 2, 1, and 0. Round the final answers to two decimals.

Method. Start at duration 2, then substitute each resulting reserve into the preceding year’s recursion. The common mistake is to use a single premium amount in all three equations.

At duration 2:

({}_2V_x^-+50)(1.05) =0.10(1000)+0.90(0)=100, \qquad {}_2V_x^-=\frac{100}{1.05}-50=45.238095\ldots.

At duration 1, the next reserve is $45.238095\ldots$:

({}_1V_x^-+40)(1.05) =100+0.90(45.238095\ldots) =100+40.714285\ldots=140.714285\ldots.

Dividing by $1.05$ gives ${}_1V_x^-+40=134.013605\ldots$, so ${}_1V_x^-=94.013605\ldots$.

At duration 0:

({}_0V_x^-+30)(1.05) =100+0.90(94.013605\ldots) =184.612244\ldots.

Dividing by $1.05$ gives ${}_0V_x^-+30=175.821185\ldots$, hence ${}_0V_x^-=145.821185\ldots$. The reserves, rounded to two decimals, are ${}_2V_x^-=\$45.24$, ${}_1V_x^-=\$94.01$, and ${}_0V_x^-=\$145.82$.

Check. Substituting each unrounded reserve into its equation reproduces the corresponding expected year-end amount: 100 at duration 2, $140.714285\ldots$ at duration 1, and $184.612244\ldots$ at duration 0. The three premiums are used at their own durations, not averaged.

Working Forward or Solving for a Premium

The recursion can also be rearranged to determine the premium consistent with specified reserves. This is useful when a reserve is known or prescribed and a duration-specific premium is unknown. Solve the same equation for $P_k$; retain the same timing and mortality assumptions.

P_k= \frac{{}_1q_{x+k}B_{k+1}+{}_1p_{x+k}\,{}_{k+1}V_x^-}{1+i} -{}_kV_x^-.

This rearrangement emphasizes that the premium depends on the reserve at the current duration and the expected benefit-plus-reserve amount one year later. It does not require the premiums at other durations to be equal.

Worked Example: Finding a Premium From a Target Reserve

A hypothetical policy has an end-of-year death benefit of $600$. The effective annual interest rate is $i=0.04$. The death probabilities are $0.10$ in the first year and $0.20$ in the second year. At duration 2, the pre-premium reserve is $100$; the desired reserve at duration 1 is $40$. The first-year premium is $P_0=80$. Find the premium $P_1$ and the initial reserve ${}_0V_x^-$. Round final dollar amounts to two decimals.

Method. First rearrange the duration-1 recursion to solve for $P_1$; then use the duration-0 recursion. The common mistake is to omit the survival-weighted next reserve or to use the first-year death probability in the second-year equation.

At duration 1, the death and survival probabilities are $0.20$ and $0.80$, respectively. Therefore:

$$ (40+P_1)(1.04) =0.20(600)+0.80(100) =120+80=200. $$

Thus $40+P_1=200/1.04=192.307692\ldots$, and $P_1=152.307692\ldots$, or $\$152.31$.

At duration 0, the death and survival probabilities are $0.10$ and $0.90$, and the premium is 80:

({}_0V_x^-+80)(1.04) =0.10(600)+0.90(40) =60+36=96.

It follows that ${}_0V_x^-+80=96/1.04=92.307692\ldots$, giving ${}_0V_x^-=12.307692\ldots$, or $\$12.31$.

Check. The duration-1 amount after premium is $40+152.307692\ldots=192.307692\ldots$; multiplied by 1.04, it equals 200. The duration-0 amount after premium is $12.307692\ldots+80=92.307692\ldots$; multiplied by 1.04, it equals 96.

Common Exam Trap: Changing Schedule Versus Level Premium

A level-premium contract uses one amount $P$ at every premium-paying duration. A duration-varying contract instead uses the stated $P_k$ at each duration. Although the recursion has the same form in both cases, the premium term must match the contract. Substituting a level amount into a nonlevel schedule changes the cash flows and therefore changes the reserves.

For example, in the two-year worked example, the premium at duration 1 is 50. Using 80 instead gives the wrong equation $({}_1V_x^-+80)(1.05)=105$, and therefore the wrong reserve ${}_1V_x^-=20$. The correct equation is $({}_1V_x^-+50)(1.05)=105$, which gives ${}_1V_x^-=50$. This is not a rounding difference: it results from modeling a different premium schedule.

Before calculating, list the premium due at each duration and identify whether the reserve is before or after that payment. Then check that each recursion uses the matching premium, one-year probabilities, benefit timing, and next-duration reserve. Negative reserves can arise for some premium schedules; they do not by themselves indicate an arithmetic error.

Key takeaway: For a pre-premium reserve, the current-duration premium enters as ${}_kV_x^-+P_k$. In a changing schedule, use $P_k$ at duration $k$; do not replace it with a premium from another year or assume a level premium.

Check Your Understanding

Use the stated annual timing convention. Unless specified otherwise, benefits are paid at year-end and reserves are immediately before the premium due at the stated duration.

A one-year term policy has $i=0.04$, death probability $0.20$, benefit $500$, premium $70$, and zero terminal reserve. Find the initial pre-premium reserve. (Answer: $\$26.15$)
A two-year term policy has $i=0.05$, death probability $0.10$ in each year, benefit $1000$, and premiums $P_0=60$, $P_1=40$. With zero terminal reserve, find ${}_1V_x^-$ and ${}_0V_x^-$. (Answer: ${}_1V_x^-=\$55.24,\ {}_0V_x^-=\$82.59$)
At duration $k$, $i=0.05$, the death probability is $0.20$, the death benefit is $400$, the next pre-premium reserve is $30$, and $P_k=50$. Find ${}_kV_x^-$. (Answer: $\$49.05$)
A two-year term policy has $i=0$, death probability $0.10$ in each year, benefit $1000$, premiums $P_0=80$ and $P_1=120$, and zero terminal reserve. Calculate ${}_1V_x^-$ and ${}_0V_x^-$. (Answer: ${}_1V_x^-=-\$20.00,\ {}_0V_x^-=\$2.00$)
At duration $k$, $i=0.05$, death probability $0.10$, benefit $1000$, next reserve $50$, and current reserve ${}_kV_x^-=25$. Find the premium $P_k$ consistent with these values. (Answer: $\$113.10$)

What You'll Learn