Duration-Dependent Premium Modifications

Why the Premium Schedule Matters

A prospective reserve is the value, at a specified duration, of future benefits less future premiums, all measured on the stated valuation basis. If the valuation premium changes with policy duration, each future premium must be valued at its own amount. Applying the current premium to every future year can materially misstate the reserve, even when the benefit assumptions are unchanged.

This tutorial considers a discrete-time, single-life contract with an annual benefit payable at the end of the year of death and premiums payable at the beginning of each policy year while the policyholder is alive. The benefit schedule and mortality assumptions are held fixed; the feature that varies is the valuation premium. These premiums are used to calculate the modified reserve and need not equal the contract premiums actually paid.

Definition: Let $P_k^M$ be the modified valuation premium assigned to policy year $k$. A duration-dependent premium modification is a rule specifying $P_k^M$ by policy year, often with a new amount beginning at a stated duration. At duration $t$, the prospective modified reserve is the expected present value of future benefits minus the expected present value of future modified valuation premiums, conditional on the policy being in force at that duration.

Timing and Prospective Reserve Formula

Use integer durations $t=0,1,\ldots,n$, where $t=0$ is issue and policy year $k$ runs from duration $k-1$ to duration $k$. Define $V_t^M$ immediately before the premium due at duration $t$. Thus the premium for policy year $t+1$ is due immediately at the valuation time. A death benefit $B_k$, if any, is paid at the end of policy year $k$.

Let $i^M$ be the effective annual interest rate on the modified valuation basis and $v^M=1/(1+i^M)$. Let $q_k^M$ be the probability of death during policy year $k$, conditional on being alive at its start, and let $p_k^M=1-q_k^M$. These are valuation assumptions. For the finite term considered here, assume premiums are payable only while the insured is alive at the start of the policy year, and there is no benefit on survival beyond the term unless explicitly stated.

Conditioning on survival to duration $t$, the probability of surviving from duration $t$ to the start of policy year $k$ is the product of the intervening $p_j^M$ values. A premium for year $k$ is paid at duration $k-1$; a benefit for death in year $k$ is paid one year later, at duration $k$. Those different payment times account for the different discount powers below.

V_t^M= \sum_{k=t+1}^{n} \left(\prod_{j=t+1}^{k-1}p_j^M\right) \left[ B_kq_k^M(v^M)^{k-t} - P_k^M(v^M)^{k-t-1} \right].

An empty product, as occurs for $k=t+1$, equals 1. The premium in the first remaining year is therefore included in full at duration $t$; it is not discounted. If no benefit is payable in a particular year, take $B_k=0$ for that year. At the end of the term, $V_n^M=0$ under the stated no-maturity-benefit assumption.

Why the Formula Holds

Each future payment contributes its amount multiplied by (1) the probability that the policy is still in force when the payment is due and (2) the discount factor from the payment date back to duration $t$. Benefits are positive liabilities and premiums reduce the reserve, which explains the subtraction.

Identify the payment year.
For a payment in policy year $k$, the policy must survive the years from $t+1$ through $k-1$. Its conditional survival probability is $\prod_{j=t+1}^{k-1}p_j^M$.

Value the benefit.
The death benefit $B_k$ is paid at the end of year $k$, so it is payable with conditional probability $q_k^M$ after surviving to that year, and discounted $k-t$ years.

Value the premium.
The premium $P_k^M$ is due at the beginning of year $k$, provided the policy has survived to that point. It is discounted $k-t-1$ years.

For a direct derivation, split the prospective value into the first remaining policy year and the years after it. At duration $t$, the premium $P_{t+1}^M$ is immediate. At the end of that year, death occurs with probability $q_{t+1}^M$, producing benefit $B_{t+1}$; survival occurs with probability $p_{t+1}^M$, after which the future reserve is $V_{t+1}^M$. Discounting the end-of-year outcomes gives the recursion

V_t^M=-P_{t+1}^M+v^M\left(q_{t+1}^M B_{t+1}+p_{t+1}^M V_{t+1}^M\right).

Repeatedly substituting this recursion through duration $n$, with $V_n^M=0$, gives the prospective sum. This also provides a useful independent check on a calculation performed by direct expected present values.

Worked Calculations

Worked Example: Three-Year Term with a Modified Premium Schedule

A hypothetical three-year term contract pays $1,000 at the end of the year of death. The valuation interest rate is $i^M=0$, and $q_k^M=0.10$ in each year. The modified valuation premiums are $P_1^M=\$59.80$, $P_2^M=\$80$, and $P_3^M=\$120$. Find $V_0^M$.

Method. Calculate the expected present values at issue, applying survival to each later-year payment. Since $i^M=0$, the discount factor is $v^M=1$.

The probability of death in year 1 is $0.10$. The probability of death in year 2 is $0.90(0.10)=0.09$, and the probability of death in year 3 is $0.90^2(0.10)=0.081$. Thus the benefit value is

1000(0.10+0.09+0.081)=1000(0.271)=\$271.00.

The first premium is immediate. The second is paid only if the policyholder survives year 1, and the third only if the policyholder survives the first two years. Therefore,

59.80+0.90(80)+0.90^2(120) =59.80+72+97.20 =\$229.00.

Subtracting the premium value from the benefit value gives

V_0^M=271.00-229.00=\boxed{\$42.00}.

Check and mistake avoided. The result is a positive net liability because the benefit value exceeds the premium value. Do not omit survival weighting on later premiums. Also keep the arithmetic exact: with a first premium of $60 instead of $59.80, the premium value would be $229.20 and the reserve would be $41.80, not $42.00.

Worked Example: Reserve at the Start of a New Premium Level

A hypothetical four-year term contract pays $4,000 at the end of the year of death. Assume $i^M=0$ and $q_k^M=0.25$ in every year. The modified valuation premiums are $100 in policy years 1 and 2, and $180 in policy years 3 and 4. Find the reserve at duration 2, immediately before the policy-year-3 premium.

Method. At duration 2, policy years 3 and 4 remain. The premium for year 3 is due immediately, while the year-4 premium is due only if the insured survives year 3. Both remaining death benefits must be weighted by the probability of reaching their policy year.

The probability of death in year 3, conditional on being in force at duration 2, is $0.25$. The probability of death in year 4 is $0.75(0.25)=0.1875$. Hence the expected value of the remaining benefits is

4000(0.25+0.75(0.25)) =4000(0.4375) =\$1{,}750.00.

The year-3 premium is immediate and the year-4 premium is paid only after survival through year 3. Its expected value is

180+0.75(180)=180+135=\$315.00.

Therefore,

V_2^M=1{,}750.00-315.00=\boxed{\$1{,}435.00}.

Check and mistake avoided. At duration 2, the year-1 and year-2 premiums and benefits are already past and do not enter the prospective reserve. The reserve is not calculated by carrying forward the original premium level: both future premiums are $180.

Worked Example: Discounting a Duration-Dependent Schedule

A hypothetical three-year term contract pays $5,000 at the end of the year of death. The valuation interest rate is $i^M=10\%$. The death probabilities are $q_2^M=0.20$ and $q_3^M=0.10$. At duration 1, immediately before the year-2 premium, the modified valuation premiums are $300 for year 2 and $700 for year 3. Find $V_1^M$.

Method. Use conditional probabilities from duration 1. The year-2 benefit is discounted one year; a year-3 benefit requires survival through year 2 and is discounted two years. Here $p_2^M=0.80$ and $v^M=1/1.10=10/11$.

The expected present value of benefits is

5000(0.20)\left(\frac{10}{11}\right) + 5000(0.80)(0.10)\left(\frac{10}{11}\right)^2 = \frac{150000}{121} =\$1{,}239.669421\ldots.

The year-2 premium is due immediately. The year-3 premium is payable one year later only if the policy survives year 2. Its expected present value is

300+(0.80)(700)\left(\frac{10}{11}\right) =300+\frac{5600}{11} =\$809.090909\ldots.

Subtracting gives $V_1^M=430.578512\ldots$, so the reserve, rounded to the nearest cent, is

\boxed{V_1^M=\$430.58}.

Check and mistake avoided. The year-3 premium is not valued as $700v^2$: it is due at the start of year 3, one year after duration 1, and is conditional on survival through year 2. Its correct contribution is $0.80(700)v$. The year-3 death benefit, in contrast, is paid at the end of year 3 and is discounted for two years from duration 1.

Applying the Schedule Reliably

A change point in the premium schedule affects only the policy years to which the new valuation amount applies. For example, if a schedule changes after duration $r$, then premiums for years $1,\ldots,r$ use the first amount and premiums for years $r+1,\ldots,n$ use the later amount. When calculating $V_t^M$, ignore premiums before duration $t$, and use the scheduled amount for each remaining year separately.

A practical calculation can be organized as a table before any arithmetic:

Policy year	Probability of reaching its start from duration $t$	Premium due at start	Death benefit value at year-end
$k$	$\prod_{j=t+1}^{k-1}p_j^M$	$P_k^M$, discounted $k-t-1$ years	$B_kq_k^M$, discounted $k-t$ years

The table makes three separate questions visible: whether the policy is in force when a payment is due, when that payment occurs, and which valuation premium applies in that policy year. It is especially useful when the premium schedule changes more than once.

Common Exam Traps

Using one premium for all future years. Apply $P_k^M$ year by year. A current valuation premium does not replace the specified future schedule.
Applying survival to the immediate premium. At duration $t$, the premium for year $t+1$ is due immediately, so its survival factor and discount factor are both 1 under the timing convention used here.
Discounting premiums like end-of-year benefits. A beginning-of-year premium for year $k$ is one year earlier than a benefit paid at the end of that year.
Including expired cash flows. A prospective reserve at duration $t$ includes future payments only; it does not include premiums already paid or benefits from earlier years.
Confusing valuation premiums with contract premiums. The schedule here is part of the modified valuation basis. If actual contractual premiums change, that is a separate cash-flow specification and must be modeled as such.

Key takeaway: For a duration-dependent premium modification, value every future premium using its own scheduled amount, its probability of being paid, and its actual payment time. Apply the same timing discipline to benefits, then subtract the premium value from the benefit value.

Check Your Understanding

For each problem, use the stated valuation basis and the same timing convention as in the worked examples.

A two-year term contract pays $2,000 at the end of the year of death. Let $i^M=0$, $q_1^M=0.10$, and $q_2^M=0.20$. The premiums are $100 and $150. Find $V_0^M$. (Answer: $325.00)
A three-year term contract pays $4,000 at the end of the year of death. Let $i^M=0$, $q_2^M=0.25$, and $q_3^M=0.10$. At duration 1, the remaining premiums are $200 for year 2 and $500 for year 3. Find $V_1^M$. (Answer: $725.00)
A two-year term contract pays $10,000 at the end of the year of death. Let $i^M=5\%$, $q_1^M=0.10$, and $q_2^M=0.20$, with premiums $500 and $600. Find $V_0^M$, rounded to the nearest cent. (Answer: $1,570.75)
A four-year term contract pays $3,000 at the end of the year of death. Let $i^M=0$ and $q_k^M=0.20$ in every year. At duration 1, premiums for years 2, 3, and 4 are $100, $200, and $400. Find $V_1^M$. (Answer: $948.00)
A three-year term contract pays $5,000 at the end of the year of death. Let $i^M=0$, $q_2^M=0.20$, and $q_3^M=0.10$. At duration 1, the year-2 premium is $250. What year-3 premium makes $V_1^M=\$800$? (Answer: $437.50)

What You'll Learn