Modified Reserve Basis Changes Mid-Contract

Why a Mid-Contract Basis Change Needs Two Reserves

A reserve basis can change while a policy remains in force. For example, an insurer may update the interest or mortality assumptions used to value future obligations. The policy’s contractual premiums and benefits do not thereby change, but the measured liability can. The key is to distinguish the reserve carried to the change date under the basis previously in force from the reserve required at that date under the new basis.

At the change date, first close the preceding period using the old basis and the cash flows that occurred during that period. Then value the remaining contract under the new basis. Comparing those two date-specific reserves identifies the immediate accounting effect of the remeasurement. It does not represent a premium or benefit paid to or by the policyholder.

Definition: Let $V_t^{\mathrm{old}}$ be the reserve immediately before a basis change at time $t$, carried forward under the basis used up to that time. Let $V_t^{\mathrm{new}}$ be the reserve at the same time for the same in-force contract, calculated using the new basis for future cash flows. Under the convention that an increase in reserves is a loss, the immediate profit effect is $$ \Pi_t=V_t^{\mathrm{old}}-V_t^{\mathrm{new}}. $$ Thus, $V_t^{\mathrm{new}}>V_t^{\mathrm{old}}$ means a reserve strengthening and an immediate loss; $V_t^{\mathrm{new}}<V_t^{\mathrm{old}}$ means a reserve release and an immediate gain.

The comparison must be made at the same duration and for the same contract conditions, such as being in force at time $t$. It should also use a consistent timing convention: for example, both reserves may be measured immediately after any premium due at time $t$. If a contractual cash flow occurs at the change date, account for that flow separately and state whether each reserve is measured before or after it.

Prospective Valuation at the Change Date

A reserve on a specified basis is a value of future net outgo, conditional on the policy being in force at the valuation time. In a prospective calculation, future benefits and expenses increase the reserve, while future premiums and other income reduce it. The basis specifies the assumptions used to value those future amounts. A change in basis may affect interest, mortality, expenses, or another valuation assumption; it does not by itself alter the contractual cash flows.

Definition: A prospective reserve at time $t$ on basis $b$, written $V_t^{(b)}$, is the present value at time $t$ of future outgo minus the present value of future income, conditional on the contract being in force at $t$. Both present values use the assumptions in basis $b$. The reserve is calculated only over the remaining contract, not by revaluing cash flows that have already occurred.

For a simple annual model, suppose a death benefit $B$ is paid at the end of a year of death, and a premium $P$ is received at the next anniversary if the policyholder survives to that anniversary. Measure the reserve immediately after the premium at time $k$, if one is due then. Let $q^{(b)}$ be the probability of death during each year on basis $b$, $p^{(b)}=1-q^{(b)}$, and $v^{(b)}=1/(1+i^{(b)})$, where $i^{(b)}$ is the effective annual interest rate on that basis. If $n$ years remain, the one-year recursion is

V_k^{(b)} = q^{(b)}v^{(b)}B + p^{(b)}v^{(b)}\left(V_{k+1}^{(b)}-P\right).

Here $V_{k+1}^{(b)}$ is measured immediately after the premium due at time $k+1$. The premium is received only if the policyholder survives the year, which is why it is inside the survival term. If no premium is due at that anniversary, set $P=0$. The terminal reserve is set by the contract’s terminal condition; for a policy with no further benefits or premiums after maturity, it is zero at maturity.

Important distinction: The recurrence carries a reserve through a period on one fixed basis. At a change date, use that recurrence to reach the date under the old basis, then calculate a separate prospective reserve for the remaining term on the new basis. Do not use the new basis to rewrite the reserve history before the change date.

Why the Reserve Recurrence Holds

The recurrence follows by conditioning on what happens during the next year. With probability $q^{(b)}$, death occurs and the insurer pays $B$ at year-end. With probability $p^{(b)}$, the policyholder survives, the premium $P$ is received at the anniversary, and the insurer then holds reserve $V_{k+1}^{(b)}$. Those year-end amounts are discounted one year to time $k$.

V_k^{(b)} = q^{(b)}v^{(b)}B + p^{(b)}v^{(b)}\left(V_{k+1}^{(b)}-P\right).

This conditional-expectation argument also explains why the calculation must respect timing. A premium due only on survival cannot be subtracted on the death path; a benefit payable at year-end must be discounted from that time. At the basis change, the same logic applies to future cash flows on the new basis, but the old-basis reserve at that date is not replaced by a reserve calculated retrospectively.

Worked Applications

Worked Example: A Mortality Strengthening at Year 1

A hypothetical three-year policy pays a death benefit of $B=\$1{,}000$ at the end of the year of death. A single premium was paid at issue, so there are no future premiums. Death during each year is conditionally independent of earlier survival, with constant annual probability $q=0.10$ on the old basis. The old effective annual interest rate is $i=0\%$. Immediately after the first policy year, the policy is in force and the mortality assumption changes to $q=0.20$; the new interest rate remains $0\%$. Calculate the old reserve carried to year 1, the new reserve, and the immediate profit effect.

Method. Calculate the old prospective reserve at issue and use the old-basis recurrence to verify the reserve carried to year 1. Then calculate the value of the two remaining years using the new mortality assumption. The common mistake avoided here is revaluing the first year as though the new assumption had applied from issue.

At issue, three annual death-benefit payments are possible. On the old basis, the probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.10)=0.09$, and $0.90^2(0.10)=0.081$, respectively. With no discounting, the reserve at issue is

V_0^{\mathrm{old}} = 1{,}000(0.10+0.09+0.081) = \$271.00.

At year 1, conditional on the policy being in force, two years remain. On the old basis, the reserve carried to that date is

V_1^{\mathrm{old}} = 1{,}000(0.10+0.90(0.10)) = \$190.00.

The recurrence confirms this roll-forward: $V_0^{\mathrm{old}}=0.10(1{,}000)+0.90V_1^{\mathrm{old}}=100+171=\$271.00$. On the new basis, the two remaining years have death probabilities $0.20$ and $0.80(0.20)=0.16$, conditional on being in force at year 1. Therefore

V_1^{\mathrm{new}} = 1{,}000(0.20+0.16) = \$360.00.

The liability increases by $\$360.00-\$190.00=\$170.00$. Under the stated convention, the immediate profit effect is $\Pi_1=190-360=-\$170.00$, an immediate loss. The check is that the new reserve equals the expected remaining death benefits under the new mortality assumption, with no discounting or future premiums.

Worked Example: A Higher Interest Basis Releases Reserve

A hypothetical two-year policy pays $\$1{,}100$ at the end of the year of death, with no survival benefit and no future premiums. Annual mortality is $q=0.10$ on both bases. The old effective annual interest rate is $i=0\%$. At the end of year 1, conditional on survival, the interest assumption changes to $i=10\%$, and one year remains. Calculate the reserve carried under the old basis, the new reserve, and the immediate profit effect.

Method. Value the one remaining death-benefit payment at the change date using each basis. The common mistake avoided is comparing a reserve at year 1 with a reserve at issue, or forgetting to discount the remaining payment under the new interest assumption.

At the start, the old-basis reserve for two years is

V_0^{\mathrm{old}} = 0.10(1{,}100)+0.90(0.10)(1{,}100) = 110+99 = \$209.00.

If the policyholder survives year 1, one year remains. The old-basis reserve carried to that date is $V_1^{\mathrm{old}}=0.10(1{,}100)=\$110.00$. On the new basis, the expected year-end payment is $0.10(1{,}100)=\$110.00$, discounted one year at $10\%$:

V_1^{\mathrm{new}} = \frac{110}{1.10} = \$100.00.

The reserve decreases by $\$10.00$, so $\Pi_1=110-100=\$10.00$: an immediate gain. The result is consistent with the assumptions: mortality is unchanged, and discounting the same expected payment at a higher interest rate lowers its present value.

Worked Example: Mortality Change With Future Premiums

A hypothetical three-year policy pays $\$1{,}000$ at the end of the year of death. A premium of $\$100$ is payable at each anniversary if the policyholder is in force, including the anniversary at maturity if the policyholder survives to it. The reserve is measured immediately after any premium due at the valuation date. There are no expenses or survival benefits, and the effective annual interest rate is $0\%$ on both bases. The old annual mortality probability is $q=0.10$. At year 1, after the premium due then has been paid and conditional on the policy being in force, mortality changes to $q=0.20$. Calculate the reserve carried to year 1, the new reserve, and the immediate profit effect.

Method. Use expected future benefits less expected future premiums, conditional on being in force at the valuation time. Premiums occur at future anniversaries only if the policyholder survives to them. The common mistake avoided is subtracting two full premiums without survival weighting.

At issue, immediately after the issue premium, the old-basis expected death benefits over the three years total

1{,}000(0.10+0.09+0.081)=\$271.00.

The future premiums at years 1, 2, and 3 are received with probabilities $1$, $0.90$, and $0.90^2=0.81$. Their expected total is $100(1+0.90+0.81)=\$271.00$. This issue-time value is not needed to remeasure at year 1, but the recurrence provides a useful roll-forward check.

At year 1, immediately after the scheduled premium, two years remain. Under the old basis, the expected benefits are $1{,}000(0.10+0.90(0.10))=\$190.00$. Future premiums at years 2 and 3 are received with probabilities $0.90$ and $0.90^2=0.81$, so their expected total is $100(0.90+0.81)=\$171.00$. Thus

V_1^{\mathrm{old}}=190-171=\$19.00.

On the new basis, the expected benefits for the remaining two years are $1{,}000(0.20+0.80(0.20))=\$360.00$. Future premiums are received with probabilities $0.80$ and $0.80^2=0.64$, giving expected premiums of $100(0.80+0.64)=\$144.00$. Therefore

V_1^{\mathrm{new}}=360-144=\$216.00.

The reserve strengthening is $\$216.00-\$19.00=\$197.00$, so the immediate profit effect is $\Pi_1=19-216=-\$197.00$, a loss. The check is that the new-basis reserve includes both increased expected death claims and reduced expected future premium income.

Applying the Change Without Misstating Profit

A disciplined calculation separates three items: the reserve brought to the change date, cash flows at that date, and the prospective reserve after remeasurement. If no contractual flow occurs at the instant of remeasurement, the immediate effect under the convention used here is simply the old reserve less the new reserve. If a premium is payable at that instant, both reserves must use the same before- or after-premium timing, and the premium must be recorded as a cash flow rather than hidden in the reserve difference.

Fix the valuation instant.
State the duration, in-force condition, and whether the reserve is measured before or after any premium or benefit due at that time.

Close the preceding period.
Carry the reserve to the change date using the old basis and the contract cash flows during the period. Do not apply the new assumptions to elapsed periods.

Value the remaining contract.
Calculate the prospective reserve at that same instant using the new basis and only future cash flows.

Interpret the difference.
Under the reserve-increase-is-a-loss convention, subtract the new reserve from the old reserve. Identify any actual cash flow separately.

The sign convention matters. Some presentations report the reserve movement $V_t^{\mathrm{new}}-V_t^{\mathrm{old}}$, which is positive for a strengthening. That quantity is not the profit effect under the convention in this tutorial; the corresponding profit effect is its negative. State the convention before interpreting a result.

This method is useful whenever an assumption or valuation rule changes during a contract. It prevents two common errors: applying a new basis to the policy’s past, and treating a change in measured liability as though cash had been received or paid. The resulting calculation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: At a mid-contract basis change, carry the existing reserve to the change date under the old basis, revalue only future obligations under the new basis, and interpret $V_t^{\mathrm{old}}-V_t^{\mathrm{new}}$ as the immediate profit effect when a reserve increase is treated as a loss.

Check Your Understanding

For each problem, assume the reserve is measured conditional on the policy being in force at the stated change date. Use the convention that an increase in reserve is an immediate loss.

A one-year death benefit of $\$2{,}000$ has one year remaining, no future premiums, and $i=0\%$. Mortality changes from $q=0.05$ to $q=0.08$. Find the new reserve and immediate profit effect. (Answer: new reserve $\$160.00$; immediate loss $\$60.00$.)
A $\$1{,}000$ death benefit has two years remaining, no future premiums, and $i=0\%$. Mortality changes from $q=0.10$ to $q=0.05$, constant in each remaining year. Find the old and new reserves and the profit effect. (Answer: old reserve $\$190.00$, new reserve $\$97.50$, immediate gain $\$92.50$.)
A $\$1{,}200$ death benefit has one year remaining and no future premiums. Mortality is $q=0.10$ on both bases. The interest assumption changes from $i=5\%$ to $i=10\%$. Find the two reserves and the profit effect, rounded to the nearest cent. (Answer: old reserve $\$114.29$, new reserve $\$109.09$, immediate gain $\$5.19$.)
A two-year policy has a $\$1{,}000$ death benefit, no future premiums, and $i=0\%$. At the change date, two years remain. Mortality changes from $q=0.08$ to $q=0.12$. Calculate the reserve under each basis and the immediate profit effect. (Answer: old reserve $\$153.60$, new reserve $\$225.60$, immediate loss $\$72.00$.)
A policy has one year remaining, a $\$1{,}000$ death benefit payable at year-end, and a $\$100$ premium due at year-end only if the policyholder survives. Let $q=0.10$ and $i=0\%$ on the old basis; on the new basis let $q=0.20$, with interest unchanged. Reserves are measured immediately after any premium due at the valuation date, so the year-end premium is a future premium from this valuation instant. Find both reserves and the immediate profit effect. (Answer: old reserve $\$10.00$, new reserve $\$120.00$, immediate loss $\$110.00$.)

What You'll Learn