Mortality Shocks in Profit Tests

Why a Mortality Shock Changes More Than Claims

A higher mortality assumption usually increases expected death claims. In a profit test, however, it also changes how many policies are expected to survive and therefore how much closing reserve remains in force. The annual profit effect is the combined result of these changes, not just the difference in claim costs.

The calculations below use a hypothetical death-benefit contract. They assume no lapses or expenses, a premium paid at the start of each policy year, and a death benefit paid at the end of the year of death. A reserve is measured immediately before the premium due at that time. This timing convention makes the opening reserve, premium, claims, and closing reserve directly reconcilable.

Definition: For a policy in force at the start of a year, let $q$ be the probability of death during that year and $p=1-q$ the probability of surviving it. Let $B$ be the end-of-year death benefit, $P$ the premium paid at the start of the year, $i$ the effective annual interest rate, $V_t$ the opening reserve immediately before that premium, and $V_{t+1}$ the reserve immediately before the next premium for a policy that survives. Expected annual profit per policy in force at the start of the year is $$ \Pi_t=(V_t+P)(1+i)-qB-pV_{t+1}. $$

The reserve $V_t$ may be calculated using an original valuation basis, while the projection uses a shocked mortality assumption. If the shock is also used to recalculate the closing reserve, then $V_{t+1}$ reflects the stressed assumptions for future years. The opening reserve is not automatically replaced: it is the amount already held at the start of the projection, unless the scenario specifically says that it is recalculated too.

Deriving the Annual Profit Equation

The formula follows from the year-end balance of the policy fund. At the start of the year, the opening reserve and premium together are $V_t+P$. They accumulate with interest to $(V_t+P)(1+i)$. A death claim of $B$ is paid with probability $q$, and a closing reserve of $V_{t+1}$ is held with probability $p$. The expected amount left after claims and survivor reserves is profit.

\begin{aligned} \text{Expected year-end assets} &= (V_t+P)(1+i),\\ \text{Expected claims} &= qB,\\ \text{Expected closing reserves} &= pV_{t+1},\\ \Pi_t &= (V_t+P)(1+i)-qB-pV_{t+1}. \end{aligned}

This is a one-year balance equation for an expectation per policy in force at the start of the year. For a portfolio of $N$ identical policies, multiply the per-policy expected profit by $N$. If mortality varies by policy or duration, calculate the relevant expected cash flows and survivor counts separately rather than applying one common factor.

A useful way to isolate a mortality shock is to compare two projections using the same opening reserve, premium, interest rate, and benefit. The shocked projection changes $q$ to $q^*$, and therefore $p$ to $p^*=1-q^*$. If the closing reserve is recalculated under the shock, it also changes to $V_{t+1}^*$. The change in profit is

\Pi_t^*-\Pi_t =-(q^*-q)B-p^*V_{t+1}^*+pV_{t+1}.

The first term is the direct change in expected claims. The remaining terms measure the change in expected closing reserves. A higher mortality rate reduces survivor exposure, but the reserve per survivor can itself rise or fall depending on the contract and the stressed future mortality basis. It is not generally correct to assume that the net reserve effect has a fixed sign.

Worked Profit-Test Calculations

Worked Example: Correcting a One-Year Portfolio Profit

A hypothetical portfolio has 1,000 identical one-year death-benefit policies. The benefit is $30,000, the premium is $650 per policy, the effective annual interest rate is 0%, and the terminal reserve is zero. Original mortality is $q_0=0.02$, while the shocked projection uses $q^*=0.023$. The opening reserve is calculated using original mortality. Find expected deaths, expected claims, and expected portfolio profit under the shock.

Method. Calculate the existing opening reserve on the original basis, then use the shocked mortality for projected claims. Since $i=0$ and the terminal reserve is zero, the reserve balance equation reduces to opening reserve plus premium less expected claims.

The original-basis opening reserve per policy is $$ V_0=q_0B-P=(0.02)(30{,}000)-650=600-650=-50. $$

Expected deaths under the shock are $1{,}000(0.023)=23$. Expected claims are $$ 23(30{,}000)=\$690{,}000.00. $$

Per policy, expected profit is $$ V_0+P-q^*B=-50+650-(0.023)(30{,}000)=-50+650-690=-90. $$ Thus expected portfolio profit is $$ 1{,}000(-90)=-\$90{,}000.00. $$

Check. On the original basis, the expected claim is $600 per policy, so $-50+650-600=0$. Under the shock, expected claims rise by $90 per policy, which takes profit from zero to $-$90$ per policy.

Common mistake. Calculating only $650-690=-40$ per policy gives $-$40,000$, but that omits the $-$50$ opening reserve per policy. The corrected calculation includes it: $-50+650-690=-90$, or $-$90,000$ for the portfolio.

Worked Example: Survival-Weighted Closing Reserves

A hypothetical two-year term contract pays $1,000 at the end of the year of death and charges a $200 premium at the start of each year while the policy is in force. The effective annual interest rate is 0%. On the original basis, the death probability is 0.20 in each year. At the start of year 1, the opening reserve is the original-basis prospective reserve. A shock changes year 1 mortality to 0.30 and year 2 mortality to 0.10. Recalculate the year 1 closing reserve using the shocked year 2 mortality, and find expected year 1 profit per policy in force.

Method. First find the opening reserve using the original basis. Then find the reserve for a year 2 survivor using the shocked year 2 mortality. Finally, weight that closing reserve by the shocked probability of surviving year 1.

On the original basis, the expected present value of benefits at time 0 is $$ (0.20)(1{,}000)+(0.80)(0.20)(1{,}000)=200+160=360. $$ The expected present value of premiums is $$ 200+(0.80)(200)=200+160=360. $$ Therefore the opening reserve is $V_0=360-360=0$.

At the start of year 2, before the year 2 premium, the shocked prospective reserve is the expected year 2 benefit less that premium: $$ V_1^*=(0.10)(1{,}000)-200=100-200=-100. $$ The year 1 shocked death probability is $q_1^*=0.30$, and the survival probability is $p_1^*=0.70$. The expected closing reserve per policy initially in force is therefore $$ (0.70)(-100)=-70. $$

Expected year 1 claims per policy initially in force are $(0.30)(1{,}000)=300$. With no interest, expected year 1 profit is $$ \Pi_0^*=(0+200)-300-(0.70)(-100)=200-300+70=-30. $$ So expected year 1 profit is $-$30$ per policy in force at the start of the year.

Check. For 1,000 policies, expected claims are $300,000 and expected survivors are 700. The closing reserve held for those survivors is $700(-100)=-$70,000$. The balance is $200,000 of premiums minus $300,000 of claims minus $-$70,000$ of reserves, giving $-$30,000$.

Common mistake. Subtracting $V_1^*=-100$ once per policy initially in force would give the wrong reserve movement. The closing reserve applies only to survivors, so its expected amount is $p_1^*V_1^*=0.70(-100)$.

Worked Example: Including Interest in a One-Year Shock

A hypothetical one-year contract has a $2,100 death benefit, a $190 premium, and an effective annual interest rate of 5%. Original mortality is $q_0=0.10$, and the shocked mortality is $q^*=0.12$. The terminal reserve is zero. The opening reserve is valued using original mortality. For a portfolio of 400 policies, calculate expected deaths, expected claims, and shocked expected profit.

Method. The opening reserve is the present value of the original expected benefit less the premium payable at the start of the year. Then accumulate opening reserves and premiums at 5%, and deduct claims under shocked mortality.

Using $v=1/1.05$, the opening reserve per policy is $$ V_0=(0.10)(2{,}100)v-190 =\frac{210}{1.05}-190 =200-190=10. $$

Expected deaths are $400(0.12)=48$. Expected claims are $$ 48(2{,}100)=\$100{,}800.00. $$ The year-end value of opening reserves and premiums for the portfolio is $$ 400(10+190)(1.05)=400(200)(1.05)=\$84{,}000.00. $$ Because the terminal reserve is zero, expected portfolio profit is $$ \$84{,}000.00-\$100{,}800.00=-\$16{,}800.00. $$

Check. Under original mortality, expected claims for the portfolio would be $400(0.10)(2{,}100)=\$84,000$, exactly the accumulated opening reserves and premiums. The shock increases expected claims by $400(0.12-0.10)(2{,}100)=\$16,800$, matching the loss.

Common mistake. Comparing premiums alone with claims, or applying interest to claims as though they were received at the start of the year, violates the stated timing. The opening reserve and premium accumulate to year-end; claims are paid at year-end.

Applying a Mortality Shock Consistently

A reliable projection separates the effects in a fixed order. Start with the policies in force at the beginning of the year. Apply the shocked mortality to that exposure to obtain expected deaths and survivors. Calculate expected claims from deaths, and calculate the expected closing reserve from survivors. Then apply the balance equation, including investment accumulation at the stated rate.

Identify the opening exposure.
Use the number of policies in force at the start of the year. Do not apply the survival probability before calculating the current year's claims.

Apply shocked mortality.
For $N$ policies, expected deaths are $Nq^*$ and expected survivors are $Np^*=N(1-q^*)$.

Set the reserve convention.
Keep the opening reserve fixed if it is already established at the shock date. Recalculate the closing reserve only if the scenario says the reserve basis is updated.

Reconcile the balance.
Accumulate opening reserves and premiums, then subtract expected claims and expected closing reserves for survivors.

For a multi-year projection, survivor exposure carries forward recursively. If $N_t$ policies are in force at the start of year $t$, then, with no other decrements, $N_{t+1}=N_t p_t$. Expected claims in that year are $N_tq_tB_t$, and expected closing reserves are $N_{t+1}V_{t+1}$. A mortality shock in an early year can therefore affect later expected claims and profits by changing the number of policies exposed in later years, even if later-year mortality assumptions are unchanged.

The equation assumes that all policies in the cohort have the same benefit, premium, mortality probability, and reserve at the relevant duration. It also assumes the stated cash-flow timing and that the reserve balance is available to support the contract. If benefits or reserves vary by policy, use the corresponding policy-level amounts or appropriate grouped exposures. If the contract has expenses, lapses, or other cash flows, include them in the balance equation with signs and timing consistent with the cash-flow model; they are not included in the examples here.

Common Exam Traps and Interpretation

Leaving out the opening reserve. Premiums less claims are not the full profit when the policy starts the year with a reserve. Include the opening balance and its investment return.
Using the wrong mortality basis. A shock to projected mortality changes expected deaths. It does not, by itself, erase an opening reserve already established on the original basis.
Forgetting survivor weighting. The closing reserve applies to survivors, so per initial policy its expected value is $pV_{t+1}$, not $V_{t+1}$.
Mixing timing conventions. In the model used here, premiums are paid at the start, claims at the end, and the closing reserve is measured just before the next premium. Changing any of these timings changes the balance equation.
Assuming a higher mortality rate always makes the reserve component a loss. The direct claim effect is adverse for a positive death benefit, but the survivor-weighted closing reserve effect depends on the reserve amount and its recalculation basis.

The practical value of this decomposition is diagnostic. It shows whether a profit change comes from additional claims, fewer surviving policies, a changed reserve per survivor, or an interaction among those items. This approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the same timing convention as in the worked examples unless the question states otherwise.

A hypothetical one-year portfolio has 600 policies, a $12,000 death benefit, a $110 premium, $i=0$, and terminal reserve zero. Original mortality is 0.01, shocked mortality is 0.012, and the opening reserve is valued on original mortality. Find expected deaths, expected claims, and expected portfolio profit. (Answer: 7.2 expected deaths; $86,400.00 expected claims; $-$14,400.00$ expected profit.)
At the start of a year, 400 policies each have an opening reserve of $20 and pay a $100 premium. The benefit is $800, $i=0$, and shocked mortality is 0.30. The closing reserve for each survivor, based on the next year's mortality, is $-$20$. Find expected claims and expected portfolio profit. (Answer: $96,000.00 expected claims; $-$42,400.00$ expected profit.)
A hypothetical one-year portfolio has 200 policies, a $5,000 benefit, a $350 premium, and $i=0.04$. Original mortality is 0.08 and shocked mortality is 0.10; terminal reserve is zero. The opening reserve is based on original mortality. Find the opening reserve per policy, expected claims, and expected portfolio profit. (Answer: opening reserve $34.62 per policy; $100,000.00 expected claims; $-$20,000.00$ expected profit.)
There are 100 policies in force at the start of a year. Each has an opening reserve of $40 and pays a $150 premium. The death benefit is $500, $i=0$, and shocked mortality is 0.20. The closing reserve remains $60 per survivor; it is not recalculated. Find expected survivors and expected profit. (Answer: 80 expected survivors; $4,200.00 expected profit.)
A portfolio of 1,000 two-year policies has no lapses. Year 1 mortality is 0.10 and year 2 mortality is 0.20, with a $1,000 benefit paid on death. Calculate expected year 1 survivors and expected year 2 claims. (Answer: 900 expected survivors; $180,000.00 expected year 2 claims.)

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