Premium Changes and the Propagation of Profit Effects

Introduction

A premium-rate change affects more than the next premium receipt. If the revised premium is expected to continue, it also changes the prospective value of future cash flows. That reserve change can offset or defer the profit effect that might appear if one looked only at the additional premium collected. In a multi-state product, the effect depends on which states pay premiums and how likely the policy is to enter each state.

The key is to trace the change in sequence: identify the affected state-contingent cash flows, recalculate the relevant reserves on the same basis, then compare profits by duration and state. This tutorial uses hypothetical products and assumptions. Its methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A premium change is a change in premium amount or rate while other specified product cash flows and assumptions are held fixed. The premium effect is the resulting change in premium cash flows and in any reserves or profits recalculated to allow for those cash flows. A premium change can affect different states differently; for example, a premium may be payable while active but not while disabled.

Trace Cash Flows Before Interpreting Profit

For a state-dependent product, write the premium rule by state and duration before calculating. Specify when premiums are due, which states pay them, and whether a rate change applies once or continues. Also state whether premiums cease on death, lapse, or another terminating event. These details determine both the expected premium cash flows and the reserve changes.

For a discrete annual model, let $V_t(s)$ be the prospective reserve at the start of year $t$ in state $s$, immediately before that year's premium and expense. Use the sign convention that a positive reserve is a liability: future benefits and expenses less future premiums. Let $P_t(s)$ and $e_t(s)$ be the premium and expense at the start of the year, and let $b_{sj}$ be the benefit paid at year-end if the transition is from state $s$ to state $j$. With annual effective interest $i$ and transition probabilities $p_{sj}$, the recursion is:

V_t(s)=e_t(s)-P_t(s)+\frac{1}{1+i} \sum_j p_{sj}\left(b_{sj}+V_{t+1}(j)\right).

This definition and recursion use the same cash-flow timing throughout: premium and expense at the start of the year, benefit at the end, and the next reserve immediately after that benefit. A terminating state has no future reserve after termination. If the product has different payment timing, the recursion must be changed to match it.

Key distinction: The additional premium is a cash inflow, while the corresponding reduction in a prospective liability is a reserve change. They are related but are not the same item. A profit calculation must use its stated reserve timing and sign convention; adding a premium increase without updating the reserve can misstate profit emergence.

Worked Example: Recalculate State Reserves and First-Year Profits

A hypothetical two-year contract has active state $A$, disabled state $D$, and death as a terminating outcome. At the start of each year, active policyholders pay a premium of \$100 and incur an expense of \$10. Disabled policyholders pay no premium and incur an expense of \$5. At year-end, the benefit is \$200 if the policy is disabled and \$1,000 if death occurs; no benefit is paid if the policy is active. From $A$, the annual transition probabilities are 0.80 to $A$, 0.15 to $D$, and 0.05 to death. From $D$, they are 0.20 to $A$, 0.70 to $D$, and 0.10 to death. The probabilities are the same in each year. Interest is $i=5\%$, so $v=1/1.05$. There is no reserve after the end of year 2. Calculate the reserves under the original premium, then find the effect of increasing the active-state premium by \$10 in each year on the year 1 conditional profits.

Method. Work backward from the terminal reserves. The transition distribution in each state is categorical with the probabilities given above, and the benefit is determined by the year-end outcome. This avoids treating the extra premium as if it were collected in every state.

At the start of year 2, the reserve after that year's cash flows is zero in every state. Thus, in state $A$:

V_1(A)=10-100+\frac{0.80(0)+0.15(200)+0.05(1000)}{1.05} =-90+\frac{80}{1.05} =-13.809524.

In state $D$, there is no premium, and the expected year-end benefit is $0.70(200)+0.10(1000)=240$. Therefore:

V_1(D)=5+\frac{240}{1.05}=233.571429.

At the start of year 1 in state $A$, the continuation values by outcome are $V_1(A)=-13.809524$, $200+V_1(D)=433.571429$, and $1000$ on death. Hence:

\begin{aligned} V_0(A) &=10-100+\frac{0.80(-13.809524)+0.15(433.571429)+0.05(1000)}{1.05}\\ &=-90+\frac{-11.047619+65.035714+50}{1.05}\\ &=-90+99.036281\\ &=9.036281. \end{aligned}

Now increase the active-state premium by \$10 in both years, leaving all other assumptions unchanged. In year 2, the extra premium reduces $V_1(A)$ by \$10, while $V_1(D)$ is unchanged because premiums are not payable in $D$. Thus $\Delta V_1(A)=-10$ and $\Delta V_1(D)=0$. At the start of year 1, the immediate premium increase is also \$10, and the expected discounted effect of the extra year 2 premium is $v(0.80)(-10)$:

\Delta V_0(A)=-10+\frac{0.80(-10)}{1.05} =-10-7.619048=-17.619048.

For a realized year 1 outcome $j$, define profit as the accumulated opening reserve plus premium less expense, accumulated for one year, less the year-end benefit and closing reserve:

\Pi_1(j)=\left(V_0(A)+P_0(A)-e_0(A)\right)(1.05)-b_{Aj}-V_1(j).

The change in this profit is therefore $(\Delta V_0+10)(1.05)-\Delta V_1(j)$. If the policy remains active, the result is $(-17.619048+10)(1.05)-(-10)=\$2.00$. If it ends the year disabled, it is $(-7.619048)(1.05)-0=-\$8.00$. If death occurs, the closing reserve is also zero, so the change is $-\$8.00$. The year-end benefit itself is unchanged, and so cancels in the profit difference.

Check the expected conditional profit change using the year 1 transition probabilities: $0.80(2.00)+0.15(-8.00)+0.05(-8.00)=1.60-1.20-0.40=\$0$. The check confirms that the prospective reserves use the same transition and interest assumptions as the profit calculation. The common mistake is to report the extra \$10 premium as a \$10 profit increase in every outcome, ignoring the changed reserve and the fact that the premium is payable only in $A$.

Why the Reserve Change Propagates

A prospective reserve is the present value of future net outgo. Under the sign convention above, raising future premiums lowers the reserve in states and durations where those premiums are expected. The change in a reserve today therefore depends not just on the immediate premium, but also on the probability of reaching future premium-paying states.

For example, in the previous calculation the reserve change at the start of year 1 includes the immediate \$10 premium increase and the discounted year 2 increase only on the 0.80 probability of being active at the next premium date. The result, $-\$17.619048$, is exactly the negative of the present value of the two expected premium increases: $10+0.80(10)/1.05=\$17.619048$. This is a direct consequence of the reserve recursion, not an additional benefit or profit.

Derivation: Subtract the reserve recursion under the original premiums from the recursion under the revised premiums. If benefits, expenses, transition probabilities, and interest are unchanged, the difference satisfies

\Delta V_t(s)=-\Delta P_t(s)+v\sum_j p_{sj}\Delta V_{t+1}(j).

The immediate term is negative because a higher premium is an income that offsets liability. The continuation term carries the reserve change forward, weighted by the probability of each next state and discounted one year. If a premium increase applies in a state that is never reached, it has no expected reserve effect. If it applies in several future durations, the recursion propagates each of those effects back to the valuation date.

Worked Example: Premium Changes in a Thiele Equation

Consider a hypothetical continuous-time product with active state $A$, disabled state $D$, and death as a terminating state. Transition intensities are constant over the instant considered: from $A$, the intensity is 0.20 per year to $D$ and 0.05 per year to death; from $D$, it is 0.10 per year to $A$ and 0.15 per year to death. The force of interest is $\delta=0.04$. A premium-rate increase of \$12 per year applies in $A$ and none applies in $D$. At the instant considered, the reserve changes are $\Delta V_A=-30$, $\Delta V_D=-10$, and $\Delta V_{\text{death}}=0$. Benefits on transitions are unchanged. Find the instantaneous derivatives of the reserve changes in $A$ and $D$.

Method. For reserve measured as future outgo less future premiums, the Thiele equation in state $s$ is $\dot V_s=\delta V_s+p_s-e_s-\sum_{j\ne s}\mu_{sj}(b_{sj}+V_j-V_s)$. Here $p_s$ and $e_s$ are premium and expense rates, $b_{sj}$ is the transition benefit, and $\mu_{sj}$ is the transition intensity. Subtracting the original and revised equations removes the unchanged expenses and benefits:

\Delta \dot V_s=\delta\Delta V_s+\Delta p_s -\sum_{j\ne s}\mu_{sj}\left(\Delta V_j-\Delta V_s\right).

The death-state reserve change is zero because death terminates the contract. In state $A$, the transition terms are $0.20(-10-(-30))=4$ and $0.05(0-(-30))=1.5$. Therefore:

\Delta\dot V_A=0.04(-30)+12-(4+1.5) =-1.2+12-5.5=5.3.

In state $D$, the transition terms are $0.10(-30-(-10))=-2$ and $0.15(0-(-10))=1.5$. Hence:

\Delta\dot V_D=0.04(-10)+0-\left(-2+1.5\right) =-0.4+0.5=0.1.

The derivatives are $+\$5.30$ per year in $A$ and $+\$0.10$ per year in $D$, at the instant specified. These are local rates of change, not reserve changes over a full year. The calculation shows why the premium increase alone does not determine the reserve derivative: the current reserve changes and transitions to states with different reserve changes also contribute.

Worked Example: NPV of State-Contingent Premium Increases

A hypothetical two-year contract receives an additional \$10 premium at time 0 if active and another additional \$10 at time 1 if active then. The policy is active with probability 1 at time 0 and has probability 0.80 of being active at time 1. Premiums are paid at those times, and the effective annual interest rate is 5%. There are no other changes in cash flows. Calculate the NPV of the additional premium receipts from time 0.

Method. Discount each expected incremental premium receipt to time 0. The time 0 increase is certain; the time 1 increase is payable only in the active state. Thus the expected time 1 receipt is $0.80(10)=\$8$, and:

\Delta NPV=10+\frac{8}{1.05} =10+7.619048 =\$17.619048.

To the nearest cent, the NPV is \$17.62. A common error is to treat the second increase as certain and calculate $10+10/1.05=\$19.52$. That calculation ignores the 0.80 probability of being active at the second premium date. In a profit test, the reserve movement must also be included according to the test's timing convention; it is not a second premium cash flow to add to this cash-flow NPV.

Separate First-Year Effects from Later Effects

A permanent premium-rate change can affect the first-year profit vector and later profit vectors differently. The first year may include the immediate premium increase, the change in the opening prospective reserve, and a closing reserve that varies by destination state. In later years, a policy may be in a state where no premium is due, or the remaining term may be shorter. Consequently, it is not generally correct to carry the first-year profit change forward as a fixed amount per policy.

A useful calculation sequence is:

Specify the premium rule.
Record the amount, payment time, duration, and states in which each premium is payable.

Revalue the reserves.
Use the same interest, transition, benefit, and expense assumptions unless the problem explicitly changes them too.

Calculate conditional profit changes.
For each possible destination state, include the premium, opening reserve, benefit, and closing reserve effects using one consistent timing convention.

Weight and discount as required.
Use transition probabilities to obtain expected profit, and discount the resulting profit signature to find its NPV.

When reserves are calculated on the same basis as the projected cash flows, a reserve change often shifts the timing of profit rather than creating additional cash. A profit-test NPV must therefore follow the stated treatment of opening reserves, closing reserves, and any initial profit. Do not add the present value of premium receipts to a profit-signature NPV without checking whether reserve releases already reflect those receipts.

Common Traps

Applying the increase in every state. If premiums stop on disability, lapse, or death, only paths in the premium-paying state generate the increase.
Using a reserve from the old premium basis. A profit calculation that changes premiums but leaves prospective reserves unchanged may produce an artificial emergence pattern.
Mixing payment times. A premium due at the start of a year and a benefit paid at year-end require different discounting and accumulation treatment.
Confusing conditional and expected profit. A conditional profit change for one transition outcome need not equal the probability-weighted expected change.
Using a discrete probability in a continuous-time equation. Thiele equations use transition intensities per unit time. They are not obtained by inserting annual transition probabilities directly.

Key takeaway: Trace a premium change through both the cash-flow projection and the reserve basis. State-specific premium rules determine which cash flows change; prospective reserves carry the expected value of future changes back through the transition model; and profit vectors show how the effect emerges by duration and outcome.

Check Your Understanding

For each problem, use the stated timing and assumptions. Amounts are hypothetical.

A contract receives an additional \$5 premium at time 0 and another \$5 at time 1 only if the policy is active then. The probability of being active at time 1 is 0.90, and $i=4\%$. Find the time-0 NPV of the additional premium receipts. (Answer: \$9.33)
An active-state premium increases by \$8 at time 0 and by \$8 at time 1 if the policy is active. The probability of being active at time 1 is 0.75, and $i=5\%$. Using future premiums as reserve offsets, find the change in the prospective liability at time 0. (Answer: $-\$13.71$)
In a discrete profit test, an outcome has opening reserve change $-\$12$, premium change $+\$7$, unchanged expenses, effective annual interest $4\%$, and closing reserve change $-\$3$. The profit change is $(\Delta V+\Delta P)(1+i)-\Delta V_{\text{close}}$. Calculate it for this outcome. (Answer: $-\$2.20$)
In a continuous-time model, $\delta=0.03$, $\Delta V_A=-20$, $\Delta V_D=-8$, and the death-state reserve change is zero. The premium-rate increase is \$6 in $A$ and zero in $D$. Intensities are $0.10$ from $A$ to $D$, $0.02$ from $A$ to death, $0.05$ from $D$ to $A$, and $0.10$ from $D$ to death. Using the subtracted Thiele equation, find $\Delta\dot V_A$ and $\Delta\dot V_D$. (Answer: $+\$3.80$ per year in $A$, $-\$0.44$ per year in $D$)
A premium increase of \$12 is payable at times 0, 1, and 2 if the policy is active. The probabilities of being active on those dates are 1, 0.60, and 0.40, respectively. With $i=3\%$, calculate the time-0 NPV. (Answer: \$23.51)

What You'll Learn

Introduction

Trace Cash Flows Before Interpreting Profit

Worked Example: Recalculate State Reserves and First-Year Profits

Worked Example: Recalculate State Reserves and First-Year Profits

Why the Reserve Change Propagates

Worked Example: Premium Changes in a Thiele Equation

Worked Example: Premium Changes in a Thiele Equation

Worked Example: NPV of State-Contingent Premium Increases

Worked Example: NPV of State-Contingent Premium Increases

Separate First-Year Effects from Later Effects

Common Traps

Check Your Understanding

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