Modified Reserves With Premium Changes

Why Premium Schedules Matter in a Modified Reserve

A policy may require different premiums at different durations: for example, an introductory premium followed by a higher renewal premium, or a premium that decreases after a specified year. A modified reserve calculation can also use a valuation premium schedule that differs from the amounts the policyholder actually pays. To calculate the reserve correctly, identify which schedule is being valued and attach each premium to its payment date.

This tutorial uses an annual, discrete model for a hypothetical term insurance policy. The death benefit is paid at the end of the policy year of death, and premiums are paid at the start of each policy year. The valuation interest and mortality assumptions are specified separately from the contractual premium schedule. All examples use those assumptions only as hypothetical inputs.

Definition: Let $n$ be the term of a policy and let $P_k$ be the premium used for valuation at time $k$, paid at the start of policy year $k+1$, for $k=0,\ldots,n-1$. Let $c_k$ be the contractual premium actually due at that time. A modified prospective reserve at time $t$ is the valuation-basis present value of future benefits minus the valuation-basis present value of future premiums $P_k$. Here, the reserve is measured immediately after the premium at time $t$ has been paid, so it includes only premiums at times strictly later than $t$.

The distinction between $c_k$ and $P_k$ is essential. The contract determines the amount the policyholder owes; the valuation basis determines the premium amounts used in the reserve calculation. A reserve is not obtained by substituting actual premiums merely because they are the amounts collected. If the valuation basis specifies a different schedule, use that schedule for the prospective valuation.

Assume throughout the derivation that the policy is in force at time $t$, annual effective interest is $i$, and mortality during each policy year is represented by conditional probabilities $q_{x+j}$ and $p_{x+j}=1-q_{x+j}$. The death benefit is a level amount $B$, payable at the end of the year of death. There is no maturity benefit, expense, lapse, or other cash flow in this model. If a contract has any such cash flow, it must be included in a valuation appropriate to that contract; it is not covered by the formula below.

Prospective Reserve Formula

Write $v=1/(1+i)$. For a policy in force at time $t$, the probability of surviving $r$ complete years is ${}_rp_{x+t}$. The probability of surviving $r-1$ years and then dying during the next year is ${}_{r-1}p_{x+t}q_{x+t+r-1}$. A benefit for death during that year is therefore discounted $r$ years from time $t$, while a premium due at time $t+r$ is discounted $r$ years and is payable only if the policyholder survives to that time.

V_t= B\sum_{r=1}^{n-t}v^r\,{}_{r-1}p_{x+t}q_{x+t+r-1} - \sum_{r=1}^{n-t-1}v^r\,{}_rp_{x+t}P_{t+r}.

The first sum includes each remaining year of death coverage, through the end of the term. The second sum includes future premium dates strictly before maturity. There is no premium at time $n$ in this model: the final premium is due at time $n-1$, at the start of the last policy year. In particular, $P_t$ does not appear because the reserve is measured just after the time-$t$ premium.

Justification: The prospective reserve is the conditional expected present value, given that the policy is in force at time $t$, of future benefits less future valuation premiums. For a benefit paid at the end of year $t+r$, the probability of payment is ${}_{r-1}p_{x+t}q_{x+t+r-1}$, and its discount factor is $v^r$. For the premium at time $t+r$, survival to that payment date has probability ${}_rp_{x+t}$, and its discount factor is also $v^r$. Summing these expected present values gives the displayed expression. This reasoning relies on the stated annual payment timing and mortality assumptions; it does not directly apply unchanged to continuous payments or benefits paid at a different time.

A useful calculation method is to make one row for every future year. Record the death probability for the benefit row, the survival probability for the premium row, the cash-flow amount, and the discount factor. This makes the two different probability weights visible rather than hiding them inside an annuity notation.

Fix the reserve date.
State whether the reserve is before or after the premium at that duration. Here it is immediately after, so the premium at time $t$ is excluded.

Write both schedules.
List contractual premiums $c_k$ and valuation premiums $P_k$ by payment time. Do not assume they match.

Value benefits and premiums separately.
Use death probabilities for benefits and survival probabilities to premium dates for premiums; discount from the stated reserve date.

Subtract and check.
Subtract the valuation-premium present value from the benefit present value, then check that the time indices and included payment dates agree with the contract term.

Worked Calculations

Worked Example: A Three-Year Term Policy With Two Different Schedules

A hypothetical three-year term policy pays $1,000 at the end of the year of death. The annual effective valuation interest rate is 10%. The conditional death probabilities by policy year are $q_{x}=0.10$, $q_{x+1}=0.20$, and $q_{x+2}=0.25$. The contractual premiums at times 0, 1, and 2 are $100, $70, and $30. The valuation premiums at those times are $120, $80, and $40. Calculate the reserve at time 0, immediately after the actual time-0 premium is paid. Round the final reserve to the nearest cent.

Method. Use the prospective formula with the valuation premiums at times 1 and 2. The time-0 valuation premium is not included because the reserve is measured after the time-0 premium. The probabilities of surviving one and two years are $0.90$ and $0.90(0.80)=0.72$.

The discount factors are $v=1/1.10=0.9090909$, $v^2=1/1.21=0.8264463$, and $v^3=1/1.331=0.7513148$. The death probabilities in policy years 1, 2, and 3 are $0.10$, $0.90(0.20)=0.18$, and $0.72(0.25)=0.18$, respectively.

The present value of benefits is:

$$ 1000(0.10)v + 1000(0.18)v^2 + 1000(0.18)v^3 =90.9091+148.7603+135.2367 =374.9061. $$

The valuation premium at time 1 is due only if the policyholder survives the first year. The valuation premium at time 2 is due only if the policyholder survives two years. Thus:

$$ 80(0.90)v+40(0.72)v^2 =65.4545+23.8017 =89.2562. $$

Therefore $V_0=374.9061-89.2562=285.6499$, so the reserve is $285.65 to the nearest cent. The contractual premiums of $70 and $30 are not used in this modified valuation: the valuation schedule specifies $80 and $40. The calculation is checked by confirming that all three benefit years and both future premium dates are included, while the already-paid time-0 premium is excluded.

Common mistake avoided: Using the actual contractual premiums would produce a different value. It would not answer the question, which specifies a valuation premium schedule that differs from the contract.

Worked Example: Valuing From a Later Duration

Use the same three-year policy, interest rate, mortality assumptions, and valuation premium schedule as in the previous example. Calculate the reserve at time 1, immediately after the actual time-1 contractual premium has been paid. Round the final reserve to the nearest cent.

Method. Re-start the prospective calculation at time 1, conditional on being in force then. The remaining death years are policy years 2 and 3. From time 1, the probability of death in policy year 2 is $0.20$; the probability of surviving that year and dying in policy year 3 is $0.80(0.25)=0.20$.

The year-2 death benefit is discounted one year, and the year-3 death benefit is discounted two years. The present value of remaining benefits is:

$$ 1000(0.20)v+1000(0.20)v^2 =181.8182+165.2893 =347.1075. $$

The only future valuation premium is $40 at time 2. It is payable if the policyholder survives from time 1 to time 2, with probability $0.80$, and is discounted one year:

$$ 40(0.80)v=29.0909. $$

Hence $V_1=347.1075-29.0909=318.0166$, or $318.02 to the nearest cent. The premium at time 1 is excluded because it has just been paid; only the time-2 valuation premium is future from this reserve date. As a check, the two remaining benefit probabilities sum to $0.20+0.20=0.40$, which is the probability of death before the end of the term conditional on being in force at time 1.

Common mistake avoided: Carrying the time-0 survival probabilities into a reserve at time 1 would calculate an issue-date probability, not a probability conditional on the policy being in force at time 1.

Worked Example: A Later Premium Reduction

A hypothetical four-year term policy pays $2,000 at the end of the year of death. The annual effective valuation interest rate is 10%, and the annual death probabilities by policy year are $0.10$, $0.15$, $0.20$, and $0.25$. Contractual premiums at times 0 through 3 are $90, $80, $25, and $10. The valuation basis instead uses premiums of $70 at times 0 and 1, then $20 at times 2 and 3. Calculate the reserve at time 1, immediately after the actual time-1 premium is paid. Round to the nearest cent.

Method. Start at time 1. There are three remaining benefit years, with conditional death probabilities $0.15$, $0.85(0.20)=0.17$, and $0.85(0.80)(0.25)=0.17$. The remaining valuation premiums are $20 at time 2 and $20 at time 3. The time-1 valuation premium is not future from this reserve date.

Using $v=1/1.10$, the benefit present value is:

$$ 2000(0.15)v+2000(0.17)v^2+2000(0.17)v^3 =272.7273+280.9917+255.4470 =809.1660. $$

For the premium at time 2, survival from time 1 to time 2 is $0.85$. For the premium at time 3, survival is $0.85(0.80)=0.68$. Therefore:

$$ 20(0.85)v+20(0.68)v^2 =15.4545+11.2397 =26.6942. $$

The reserve is $809.1660-26.6942=782.4718$, or $782.47 to the nearest cent. The calculation uses the lower valuation premium schedule after time 1, even though the contractual schedule also changes and has different amounts. The check on the mortality weights is $0.15+0.17+0.17=0.49$, the conditional probability of death during the three remaining years.

Common mistake avoided: Using $25 and $10 because they are the contractual premiums would mix the actual schedule with the valuation basis. The prospective reserve requires the stated valuation amounts of $20 and $20.

Interpretation and Exam Checks

A duration-dependent valuation premium changes the reserve through the present value of future premium income. A larger valuation premium, all else equal, reduces the prospective reserve; a smaller valuation premium increases it. This effect depends on both the amount and the probability-weighted payment date. A premium that is large but due far in the future may have less present value than a smaller premium due soon.

The most reliable checks are structural. First, confirm the reserve timing: “just after the premium” excludes that date’s premium. Second, confirm that each premium is weighted by survival to its payment date, not by the probability of dying in the corresponding year. Third, confirm that benefits use the probability of surviving to the year and then dying in it. Finally, use the valuation schedule consistently rather than blending it with contractual amounts.

Key takeaway: A modified prospective reserve is built from future cash flows on the valuation basis. When premiums vary by duration, list each future premium with its date, valuation amount, survival probability, and discount factor before subtracting its present value from the benefit present value.

Check Your Understanding

For each question, use the stated reserve timing and valuation premium schedule. Unless stated otherwise, benefits are paid at the end of the year of death, premiums are paid at the start of a policy year, and there are no expenses or other benefits.

A two-year term policy has a $1,000 death benefit, zero interest, and death probabilities $q_x=0.10$ and $q_{x+1}=0.20$. At time 0, immediately after the initial premium, the valuation premium at time 1 is $50. Calculate $V_0$. (Answer: $235.00)
A three-year term policy has a $1,000 benefit, zero interest, and death probabilities $0.10$, $0.20$, and $0.25$. At time 0 after the initial premium, the valuation premiums at times 1 and 2 are $100 and $50. Calculate $V_0$. (Answer: $334.00)
For a three-year term policy, $i=10\%$, the death probabilities are $0.10$, $0.20$, and $0.25$, and the benefit is $1,000. At time 1 after the premium, the only future valuation premium is $40 at time 2. Calculate $V_1$, to the nearest cent. (Answer: $318.02)
A three-year term policy has a $1,000 benefit, zero interest, and death probability $0.10$ in each year. At time 0 after the initial premium, valuation premiums of $60 and $30 are due at times 1 and 2. Calculate $V_0$. (Answer: $192.70)
A three-year term policy has a $1,000 benefit, zero interest, and death probabilities $0.10$, $0.20$, and $0.25$. At time 0 after the initial premium, the valuation premiums due at times 1 and 2 are $80 and $40, while the contractual premiums are $70 and $30. Calculate the valuation reserve and the amount by which it differs from a reserve incorrectly calculated using the contractual schedule. (Answer: valuation reserve $359.20; the reserve using contractual premiums would be $375.40, which is $16.20 higher.)

What You'll Learn