Premium and Benefit Basis Sensitivity

Why Contract Changes Affect Both Cash Flows and Reserves

A contractual premium or benefit change alters the cash flows in a profit test. It can also alter the prospective reserve, because that reserve represents the value of future contractual benefits and expenses less future contractual premiums. A profit comparison that updates one but not the other can therefore give a misleading picture of profit emergence.

The essential distinction is between changing the contract and changing the assumptions used to value it. A revised premium schedule changes future premium cash flows, and a revised benefit changes future benefit cash flows. Neither change, by itself, changes the mortality or interest assumptions used to calculate a reserve. The revised reserve must be calculated for the revised contract on the specified valuation basis.

Model and convention: Consider a hypothetical annual term insurance contract with $n$ policy years. A premium $P_t$ is due at the start of year $t$, provided the policy is in force. A death benefit $B_t$ is paid at the end of that year if death occurs. There are no expenses in the examples unless stated. The valuation basis has annual interest rate $j$, death probability $q^v$, and survival probability $p^v=1-q^v$. The reserve $V_t$ is measured immediately before the premium due at time $t$; $V_n=0$. Actual profit calculations may use different assumptions, denoted $i^a$, $q^a$, and $p^a=1-q^a$.

Prospective Reserve and Reserve-Adjusted Profit

At time $t$, the prospective reserve is the present value, on the valuation basis, of future benefits less the present value of future premiums. The premium due immediately at time $t$ is included as a future premium. With the timing conventions above, the reserve recursion is

V_t=-P_t+\frac{q^v B_t+p^v V_{t+1}}{1+j}.

The term $q^vB_t$ is the expected end-of-year death benefit on the valuation basis. The term $p^vV_{t+1}$ is the expected reserve at the end of the year for policies surviving to time $t+1$. Both are discounted one year. The start-of-year premium is not discounted, so it enters as $-P_t$.

For a profit test using actual expected experience, let $E_t$ be an expense paid at the start of year $t$. Conditional on a policy being in force at the start of the year, the expected end-of-year reserve-adjusted profit is

\pi_t=(V_t+P_t-E_t)(1+i^a)-q^aB_t-p^aV_{t+1}.

The opening reserve, premium, and expense form the amount invested through the year. At year-end, expected death claims and the reserve held for surviving policies are deducted. The expected profit signature per policy issued at time 0 weights this conditional profit by the probability of remaining in force: its year-$t$ entry is ${}_tp_x^a\pi_t$, where ${}_tp_x^a$ is the actual probability of survival to time $t$. If actual and valuation assumptions differ, a reserve change alone does not determine the change in profit; the actual cash flows and actual assumptions also matter.

How a Contractual Change Revises the Reserve

Let $\Delta P_t$ and $\Delta B_t$ be changes to the premium and benefit in year $t$. If the valuation assumptions stay fixed, the reserve change $\Delta V_t$ satisfies

\Delta V_t=-\Delta P_t+ \frac{q^v\Delta B_t+p^v\Delta V_{t+1}}{1+j}, \qquad \Delta V_n=0.

This follows by writing the prospective reserve recursion once for the revised contract and once for the original contract, then subtracting. In particular, a higher premium reduces the reserve, all else equal, because it increases future income; a higher benefit increases it, because it increases future outgo. The effects can occur together and can offset.

For a level premium increase $\Delta P$ due at each remaining anniversary, with no benefit change, the reserve at time $t$ falls by the valuation-basis present value of those premiums:

\Delta V_t=-\Delta P\sum_{k=0}^{n-t-1} {}_kp^v v_j^k, \qquad v_j=\frac{1}{1+j}.

The sum starts at $k=0$, since the premium due at time $t$ is immediate under the stated convention. For benefit changes, the recursive expression, rather than a premium-annuity expression, is the direct way to calculate the reserve impact.

Worked Example: Raising Premiums on a Two-Year Contract

A hypothetical two-year term contract has annual premium $P=100$, annual death benefit $B=1{,}000$, valuation interest $j=10\%$, and valuation death probability $q^v=0.10$ in each year. Thus $p^v=0.90$. Premiums are due at the start of each year, and death benefits at year-end. The annual premium is increased to $110$, with no benefit change. Recalculate the reserves and check the expected profit signature when actual assumptions equal the valuation assumptions.

Method. Work backward from $V_2=0$, first calculating the original and revised reserves. Then use the reserve-adjusted profit formula. This method avoids treating the premium increase as a change in mortality or interest.

For the original contract, the time-1 reserve is

V_1=-100+\frac{0.10(1{,}000)+0.90(0)}{1.10} =-100+\frac{100}{1.10} =-\frac{100}{11} =-9.0909\ldots.

The time-0 reserve is

V_0=-100+\frac{100+0.90(-100/11)}{1.10} =-100+\frac{1{,}010/11}{1.10} =-\frac{2{,}000}{121} =-16.5289\ldots.

For the revised premium, the time-1 reserve becomes

V_1^{\mathrm{new}}=-110+\frac{100}{1.10} =-\frac{210}{11} =-19.0909\ldots.

At time 0, the revised reserve is

V_0^{\mathrm{new}} =-110+\frac{100+0.90(-210/11)}{1.10} =-\frac{4{,}200}{121} =-34.7107\ldots.

Thus the reserve changes are $\Delta V_1=-10$ and $\Delta V_0=-2{,}200/121=-18.1818\ldots$. The time-0 reserve falls by more than $10$ because the higher premium is due at time 0 and another higher premium is due at time 1 if the policyholder survives.

Using the revised contract, the conditional expected profit in year 0 is

\pi_0=(V_0^{\mathrm{new}}+110)(1.10)-0.10(1{,}000)-0.90V_1^{\mathrm{new}} =\frac{911}{11}-100+\frac{189}{11}=0.

In year 1,

\pi_1=(V_1^{\mathrm{new}}+110)(1.10)-0.10(1{,}000) =100-100=0.

The profit signature is therefore $(0,0)$, also after survival weighting. This is expected because actual and valuation assumptions coincide and the reserves are calculated for the revised contract. The reserve changes offset the revised cash flows in the reserve-adjusted profit calculation.

Worked Example: Increasing the Death Benefit

Use the same two-year contract, valuation assumptions, premium of $100$, and original benefit of $1{,}000$ in each year. Now increase the death benefit to $1{,}100$ in each year while leaving the premium unchanged. Find the revised reserves and verify the profit signature under the matching valuation and actual assumptions.

Method. Recalculate the prospective reserves from the end of the term backward. The common mistake this avoids is counting the higher claim cash flow but retaining reserves based on the old, lower benefit.

At time 1, the revised reserve is

V_1^{\mathrm{new}}=-100+\frac{0.10(1{,}100)}{1.10}=0.

At time 0,

V_0^{\mathrm{new}} =-100+\frac{0.10(1{,}100)+0.90(0)}{1.10} =-100+\frac{110}{1.10}=0.

For comparison, the original reserves were $V_1=-100/11$ and $V_0=-2{,}000/121$. The reserve increases are therefore $100/11=9.0909\ldots$ at time 1 and $2{,}000/121=16.5289\ldots$ at time 0. The earlier reserve also reflects the higher expected benefit in the first year and the higher reserve expected for survivors.

The revised-contract profit in year 0 is

\pi_0=(0+100)(1.10)-0.10(1{,}100)-0.90(0)=110-110=0.

In year 1 it is

\pi_1=(0+100)(1.10)-0.10(1{,}100)=110-110=0.

Again the profit signature is $(0,0)$ when the valuation and actual assumptions match. The higher benefit increases both expected claim outgo and the reserves supporting future benefits; using an old reserve with the new claim would break this reconciliation.

When Actual and Valuation Assumptions Differ

A contract revision need not produce zero expected profit. If actual expected experience differs from the valuation basis, the revised contract is still valued on the valuation basis, while the profit test uses actual expected experience. The profit then reflects both the revised contractual amounts and the difference between actual experience and the reserve assumptions.

Worked Example: Revised Premium and Benefit with Different Actual Mortality

A hypothetical one-year term contract has original premium $20$, death benefit $200$, valuation death probability $q^v=0.10$, and valuation interest $j=5\%$. Actual expected death probability is $q^a=0.08$, and actual interest is $i^a=4\%$. There are no expenses. The revised contract has premium $22$ and death benefit $240$. Calculate the old and revised reserves and expected profits, then reconcile the profit change.

Method. Calculate each reserve using valuation assumptions, then calculate each profit using actual assumptions. This keeps the contract amounts, valuation basis, and actual experience basis distinct.

The original reserve, with terminal reserve zero, is

V_0=-20+\frac{0.10(200)}{1.05} =-20+\frac{400}{21} =-\frac{20}{21} =-0.9524\ldots.

The original expected profit is

\pi_0=(V_0+20)(1.04)-0.08(200) =\frac{400}{21}\frac{26}{25}-16 =\frac{80}{21} =3.8095\ldots.

For the revised contract, the reserve is

V_0^{\mathrm{new}}=-22+\frac{0.10(240)}{1.05} =-22+\frac{160}{7} =\frac{6}{7} =0.8571\ldots.

The revised expected profit is

\pi_0^{\mathrm{new}} =(V_0^{\mathrm{new}}+22)(1.04)-0.08(240) =\frac{160}{7}\frac{26}{25}-19.2 =\frac{32}{7} =4.5714\ldots.

The reserve increases by $6/7-(-20/21)=38/21=1.8095\ldots$, while expected profit increases by $32/7-80/21=16/21=0.7619\ldots$. Directly, the profit change is

\Delta\pi_0 =(\Delta V_0+\Delta P)(1.04)-q^a\Delta B =\left(\frac{38}{21}+2\right)(1.04)-0.08(40) =\frac{16}{21} =0.7619\ldots.

The profit change is not just the premium increase less expected additional claims. The reserve also changes, and the accumulation rate on the reserve-plus-premium amount is the actual interest rate. Rounding only the final monetary answers to the nearest cent gives an original reserve of $-\$0.95$, a revised reserve of $\$0.86$, an original expected profit of $\$3.81$, a revised expected profit of $\$4.57$, and a profit increase of $\$0.76$.

Common Mistakes and Practical Checks

A useful discipline is to identify, separately, the contractual cash flows, the valuation basis, and the actual assumptions used in the profit projection. A premium or benefit revision changes the first item. It changes the reserve only after the prospective valuation is recalculated; it does not automatically change the other two.

Omitting future premiums from a reserve change. If the revised premium is due at several future anniversaries, include each premium that remains payable while the policy is in force. The time-0 change is not generally just the change to the immediate premium.
Using the wrong timing for the current premium. Under the convention used here, the premium due at time $t$ is immediate and is included in $V_t$ without discount. If a problem specifies a different premium timing, the recursion must change accordingly.
Updating claims but not reserves. A benefit increase changes the expected claim amount and the prospective value of future benefits. Use the revised benefit schedule when recalculating reserves.
Mixing actual and valuation assumptions. Use valuation mortality and interest in $V_t$, and actual mortality and interest in $\pi_t$. Substituting one basis for the other changes the calculation.
Confusing conditional profit with profit signature. The formula for $\pi_t$ is conditional on being in force at time $t$. To obtain the expected profit per policy issued, weight it by the probability of reaching that year in force.

For a hypothetical product analysis, documenting the effective date of a contractual change is also important. If a premium increase applies only from the next anniversary, it must not be included as a premium already received. Similarly, a benefit change may apply only to future coverage periods. Revalue only the contractual cash flows that are actually changed.

Key takeaway: Recalculate the prospective reserve for the revised contractual premium and benefit schedule, using the stated valuation basis. Then project profit using the revised cash flows, the revised reserves, and the stated actual assumptions. This separation is central to sensitivity analysis aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the timing conventions stated in the question and keep valuation assumptions separate from actual assumptions where relevant.

A one-year term contract has valuation death probability $0.08$, valuation interest $5\%$, and premium $30$ due at the start of the year. The death benefit increases from $300$ to $350$. Find the change in the time-0 prospective reserve. (Answer: reserve increases by $\$3.81$)
A two-year contract has valuation survival probability $0.80$ and valuation interest $4\%$. The premium due at the start of each remaining year increases by $5$, with no benefit change. Find the reserve change at time 0. (Answer: $-\$8.85$)
A one-year contract has premium $25$, benefit $200$, valuation death probability $0.10$, and valuation interest $5\%$. Actual death probability is $0.08$ and actual interest is $3\%$. With no expenses, calculate the time-0 reserve and expected profit. (Answer: reserve $-\$5.95$; expected profit $\$3.62$)
A two-year contract has valuation death probability $0.10$, survival probability $0.90$, and valuation interest $10\%$. The death benefit increases by $50$ in each year, with premiums unchanged. Find the time-0 reserve increase. (Answer: $\$8.26$)

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