Mortality Assumption Changes in Profit and Reserve Tests

Why Mortality Changes Must Be Traced Through the Whole Test

A change in mortality affects more than the expected number of death claims. It also changes how many policies remain in force to pay later premiums or receive a survival benefit. If reserves are recalculated, the change can alter profit at a different time from the change in claims. A reliable analysis therefore keeps three items distinct: the mortality assumption used to project experience, the mortality assumption used to calculate reserves, and the timing of cash flows.

The examples use a hypothetical two-year contract with a death benefit and a maturity benefit. The contract pays a premium at the start of each year while the policy is in force. A death benefit is paid at the end of the year of death; a maturity benefit is paid at the end of year 2 if the insured survives. There are no expenses or lapses. Unless stated otherwise, the annual effective interest rate is $i=5\%$, and all amounts are in dollars.

Definition: In a two-state model, the insured is either alive or dead. For year $t$, let $q_t$ be the conditional probability of death during that year given survival to its start, and let $p_t=1-q_t$. If the force of mortality is constant at $\mu_t$ during that year, then $q_t=1-e^{-\mu_t}$ and $p_t=e^{-\mu_t}$. The dead state is absorbing: once entered, it cannot be left.

A mortality-intensity change must be translated into annual transition probabilities before projecting cash flows. For example, changing a constant force from $\mu=-\ln(0.9)$ to $\mu'=-\ln(0.8)$ changes the annual death probability from $0.10$ to $0.20$, and the survival probability from $0.90$ to $0.80$. The change is applied conditionally to those alive at the start of the relevant year, not to the original population in every year.

Occupancy, Claims, and Benefit Timing

Let $N_t$ be the expected number alive at the start of year $t$. For a cohort with no entry or exit other than death, $N_{t+1}=N_t p_t$, and the expected number of deaths during year $t$ is $N_tq_t$. If death claims are $B$ and the maturity benefit is $M$, expected payments during that year are $N_tq_tB$. Expected maturity payments are based on the number surviving to maturity, not on the number initially insured.

Worked Example: Reprojecting Occupancy and Claims

A hypothetical cohort has 1,000 policies. The original annual death probability is $0.10$ in each of two years. A new mortality assumption changes year 1 only to $q_1=0.20$; year 2 remains at $q_2=0.10$. Each death benefit is $1,000 and the year-2 maturity benefit is $1,100. Find the expected occupancy and total undiscounted benefits under each assumption. The corresponding constant forces are $-\ln(0.90)=0.1054$ and $-\ln(0.80)=0.2231$ per year.

Original assumption. At the start of year 1, 1,000 are alive. Year-1 deaths are $1{,}000(0.10)=100$, leaving $1{,}000(0.90)=900$ alive for year 2. Year-2 deaths are $900(0.10)=90$, leaving $900(0.90)=810$ to receive maturity benefits.

Expected death claims are $100($1{,}000)=$100{,}000$ in year 1 and $90($1{,}000)=$90{,}000$ in year 2. Expected maturity benefits are $810($1{,}100)=$891{,}000$. Total expected undiscounted benefits are $$100{,}000+$90{,}000+$891{,}000=$1{,}081{,}000$.

Changed assumption. Year-1 deaths are $1{,}000(0.20)=200$, leaving $1{,}000(0.80)=800$ alive at the start of year 2. Year-2 deaths are $800(0.10)=80$, leaving $800(0.90)=720$ to receive maturity benefits.

Expected death claims are $200($1{,}000)=$200{,}000$ in year 1 and $80($1{,}000)=$80{,}000$ in year 2. Expected maturity benefits are $720($1{,}100)=$792{,}000$. The new total is $$200{,}000+$80{,}000+$792{,}000=$1{,}072{,}000$, a decrease of $$9{,}000$ from the original total. The larger first-year death claim is more than offset by fewer later claims and fewer maturity benefits.

Check. The changed projection has $200+80+720=1{,}000$ policies accounted for across death in year 1, death in year 2, and survival to maturity.

Prospective Reserves and the Assumption Basis

For the reserve calculations below, $V_t$ is the prospective reserve immediately before the premium due at time $t$, conditional on being alive then. Premiums are paid at times 0 and 1 if the policy is alive; $V_2=0$ immediately before any payment at maturity. A reserve is calculated from the expected present value of future benefits less the expected present value of future premiums, using the specified valuation mortality and interest assumptions.

Theorem: With annual premium $P$, death benefit $B$, annual death probability $q_t$, survival probability $p_t=1-q_t$, and annual effective interest rate $i$, a nonterminal reserve satisfies $$ V_t=-P+v\bigl(q_tB+p_tV_{t+1}\bigr),\qquad v=\frac{1}{1+i}. $$ In the final year, when survival leads to maturity benefit $M$ rather than a continuing reserve, the formula is $$ V_1=-P+v\bigl(q_1B+p_1M\bigr). $$

These equations follow by conditioning on the events during the coming year. The premium is received immediately, so it reduces the net future liability by $P$ without discounting. At year-end, death occurs with probability $q_t$ and produces payment $B$; survival occurs with probability $p_t$ and leaves reserve $V_{t+1}$ (or produces $M$ in the final year). The expected year-end value is discounted by $v$. This recursion assumes annual premiums and benefits at the stated times, the stated interest rate, and the specified conditional mortality probabilities. It does not apply unchanged if payment timing, benefit amounts, or transition states differ.

Worked Example: Recalculating Reserves After a Mortality-Basis Change

Use the two-year contract with $P=$100$, $B=$1{,}000$, $M=$1{,}100$, and $i=5\%$. Under the original valuation basis, $q_1=q_2=0.10$. Recalculate reserves if the valuation basis changes to $q_1=q_2=0.20$, while keeping the premium and benefits unchanged. The corresponding forces are $0.1054$ and $0.2231$ per year.

Original basis. Here $v=1/1.05=0.95238095$. At time 1, $$ V_1=-100+\frac{0.10(1{,}000)+0.90(1{,}100)}{1.05} =-100+\frac{1{,}090}{1.05} =$938.0952. $$ At time 0, $$ V_0=-100+\frac{0.10(1{,}000)+0.90(938.0952)}{1.05} =-100+\frac{944.2857}{1.05} =$799.3197. $$

Changed basis. At time 1, $$ V_1'=-100+\frac{0.20(1{,}000)+0.80(1{,}100)}{1.05} =-100+\frac{1{,}080}{1.05} =$928.5714. $$ At time 0, $$ V_0'=-100+\frac{0.20(1{,}000)+0.80(928.5714)}{1.05} =-100+\frac{942.8571}{1.05} =$797.9592. $$

The changes are $V_1'-V_1=$928.5714-$938.0952=-$9.5238$ and $V_0'-V_0=$797.9592-$799.3197=-$1.3605$. The higher death probability reduces these reserves because the maturity benefit is larger than the death benefit, and more policies are expected to receive the smaller benefit instead of the larger one.

Check. The reserve recursion is satisfied on the changed basis: $-100+(200+0.8(928.5714))/1.05=$797.9592$, agreeing with $V_0'$ to the displayed precision.

Transition Profits, Annual Results, and NPV

A profit test must say whether reserves are held on the original valuation basis or recalculated on the changed basis. These answer different questions. Holding reserves fixed while changing actual mortality measures the effect of an experience deviation against the existing reserve basis. Recalculating reserves measures the liability under a new valuation assumption. Do not mix the two within one calculation.

Under the convention used here, the profit at year-end equals accumulated funds, including the reserve held at the start and the premium received, less the claim or reserve carried forward. Conditional on being alive at time $t$, death and survival transition profits are

\Pi_t^{D}=(V_t+P)(1+i)-B,\qquad \Pi_t^{S}=(V_t+P)(1+i)-V_{t+1}.

In the final year, replace $V_{t+1}$ in the survival expression with $M$. Expected profit per policy in force at the start of the year is $q_t\Pi_t^D+p_t\Pi_t^S$. To obtain expected profit for the original cohort, multiply by the probability of being alive at the start of that year. Discount each year-end expected profit to time 0 to calculate NPV.

The formulas are a balance-sheet identity: accumulated opening reserve and premium are the funds available at year-end; the transition determines either the benefit paid or the liability carried forward. The result depends on the stated sign convention and payment timing. In particular, a death-state profit is not the same as the expected profit across both transitions.

Worked Example: Profit Signature and NPV Under Higher Experience Mortality

Use $P=$100$, $B=$1{,}000$, $M=$1{,}100$, and $i=5\%$. Keep the original valuation reserves $V_0=$799.3197$, $V_1=$938.0952$, and $V_2=0$, but change experience mortality to $q_1=q_2=0.20$, so $p_1=p_2=0.80$. Calculate transition profits, expected annual profits for one policy issued, and NPV. This is an experience test against the original reserve basis, not a recalculation of reserves.

Year 1 transition profits. Funds accumulated to year-end are $(799.3197+100)(1.05)=$944.2857$. If death occurs, profit is $$944.2857-$1{,}000=-$55.7143$. If the policy survives, profit is $$944.2857-$938.0952=$6.1905$. Weighting by the changed transition probabilities gives expected profit conditional on being in force: $$ 0.20(-55.7143)+0.80(6.1905) =-11.1429+4.9524=-$6.1905. $$ All 1,000? No cohort multiplier is needed: this result is for one policy issued. The expected year-1 profit for that policy is $-$6.1905$.

Year 2 transition profits. Funds accumulated for a policy alive at time 1 are $(938.0952+100)(1.05)=$1{,}090.0000$. Death profit is $$1{,}090-$1{,}000=$90$; survival-to-maturity profit is $$1{,}090-$1{,}100=-$10$. Conditional expected profit is $0.20($90)+0.80(-$10)=$18-$8=$10$. Only $0.80$ of the original policies are expected to reach year 2, so expected year-2 profit per policy issued is $0.80($10)=$8$.

NPV. Discount the year-end results at $5\%$: $$ \operatorname{NPV}=\frac{-6.1905}{1.05}+\frac{8}{1.05^2} =-5.8957+7.2562=$1.3605. $$ The profit signature per policy issued is $(- $6.1905,\; $8.0000)$, and its NPV is positive. The result is checked by retaining the year-2 survival probability of $0.80$ when aggregating: using the conditional year-2 profit of $10 directly as the cohort profit would overstate that year's contribution.

Common Exam Traps

Applying a conditional death probability to the original cohort every year. In the first worked example, the changed year-2 deaths are $800(0.10)=80$, not $1{,}000(0.10)=100$. Year-2 mortality applies only to those who survived year 1.
Changing experience mortality but silently changing reserves too. An experience profit test can hold the valuation reserves fixed. If the reserve basis is also changed, calculate that separately and identify the new basis.
Confusing conditional and cohort profit. Year-2 expected profit conditional on being in force is $10 in the third example. The expected year-2 profit per policy issued is $0.80($10)=$8$.
Discounting the wrong amount or at the wrong time. The profit is measured at year-end, so discount it by $v^t$. Do not discount a conditional profit and then omit the probability of reaching that year.

Key takeaway: Recalculate mortality-dependent occupancy first, then claims and survival benefits. State explicitly whether reserves use the old or new mortality basis. Build transition profits from that reserve basis, weight them by the appropriate state probabilities, and discount the resulting annual profit signature.

Check Your Understanding

For each problem, state the conditional probabilities and the basis being used before calculating.

A cohort of 500 has annual death probability $0.12$ in year 1 and $0.08$ in year 2. How many are expected to be alive at the start of year 2, to die in year 2, and to survive both years? (Answer: 440 alive at the start of year 2; 35.2 expected year-2 deaths; 404.8 expected to survive both years.)
A constant force of mortality is $0.15$ per year. Find the one-year death and survival probabilities. (Answer: $q=0.1393$, $p=0.8607$, to four decimals.)
At the start of a year, $V=$420$, $P=$80$, and $i=4\%$. A death claim of $600 is paid at year-end; a survivor carries reserve $V_1=$450$. If $q=0.15$, find the death profit, survival profit, and expected profit per policy in force. (Answer: death profit $-$80$; survival profit $70; expected profit $47.50.)
For the two-year contract with $P=$100$, $B=$1{,}000$, $M=$1{,}100$, and $i=5\%$, the original mortality probabilities are $0.10$ in both years. If only the year-1 valuation probability changes to $0.20$, while year 2 remains $0.10$, find the revised $V_0$ and its change from the original $V_0$. (Answer: revised $V_0=$805.22$; increase of $5.90.)
Use the contract and original reserves in the worked profit example. If experience mortality changes to $q_1=0.20$ but remains $q_2=0.10$, calculate expected year-1 profit per policy issued and its time-0 present value. (Answer: year-1 expected profit $-$6.19$; present value $-$5.90$.)
For the same contract and original reserves, suppose experience mortality is $0.20$ in both years. What is the year-2 expected profit per policy issued and the NPV over both years? (Answer: year-2 expected profit $8.00; NPV $1.36.)

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