Mortality Improvement and Cohort Trend Stresses

Why Mortality Improvement Requires a Cohort View

A mortality assumption can change over calendar time even when the underlying table is indexed by attained age. A person who reaches a given age in a later year may therefore have a different one-year probability of death from someone who reached that age earlier. This matters when projecting long-term contracts: the relevant mortality rates follow the policyholder along a path through both attained ages and calendar years.

A mortality improvement rate describes a proportional reduction in a mortality measure over time. It is not the same as a one-time mortality level shock. For this tutorial, improvement is applied multiplicatively to annual death probabilities. Other models may apply improvement to central death rates or forces of mortality, so the measure being stressed must always be identified before calculating.

Definition: Let $q_x^{\mathrm{base}}$ be the annual probability of death at attained age $x$ in a stated base calendar year $T_0$. Under scenario $s$, let $m_{x,j}^{(s)}$ be the fractional mortality improvement from calendar year $j$ to $j+1$ at age $x$. A specified projected rate is $$ q_{x,t}^{(s)}=q_x^{\mathrm{base}}\prod_{j=T_0}^{t-1}\left(1-m_{x,j}^{(s)}\right). $$ For $t=T_0$, the product is empty and equals 1. This convention assumes the improvement rates apply to annual probabilities of death, and the resulting probabilities must remain between 0 and 1.

If improvement is constant at rate $m$, the expression simplifies to $q_{x,t}=q_x^{\mathrm{base}}(1-m)^{t-T_0}$. The exponent counts calendar-year steps from the base year, not the policy duration by itself. For a cohort issued at age $x_0$ in calendar year $c$, the rate at duration $k$ is the rate for attained age $x_0+k$ and calendar year $c+k$. Thus, the cohort follows a diagonal through an age-by-calendar-year table.

A mortality improvement stress changes the assumed improvement rates and projects the consequences. A valid comparison holds other assumptions fixed, such as interest, benefits, expenses, and premium terms. The direction of the financial effect depends on the contract: lower mortality generally reduces death claims on life insurance but may increase expected payments on a life annuity.

Projecting Claims Along a Cohort

Suppose a death benefit $B_k$ is paid at the end of policy year $k$ if death occurs during that year. Write $q_k^{(s)}$ for the projected mortality rate at the cohort’s attained age and calendar year in that duration. If mortality follows the specified annual probabilities and there are no other decrements, the probability of death in year $k$ is the probability of surviving the preceding $k-1$ years multiplied by the year-$k$ death probability. With effective annual interest rate $i$, $v=1/(1+i)$, so the expected present value of claims is

\operatorname{EPV}^{(s)}(\text{claims}) =\sum_{k=1}^{n}v^k B_k \left[\prod_{r=1}^{k-1}(1-q_r^{(s)})\right]q_k^{(s)}.

This expression assumes annual benefits paid at year-end of death, deterministic interest, and no lapses or other causes of exit. If benefits are paid immediately on death, or other decrements affect the number exposed to mortality, the cash-flow timing or probability model must be adjusted.

Worked Example: Comparing Two Issue Cohorts’ Death Claims

A hypothetical two-year term policy pays $100,000 at the end of the year of death. The effective annual interest rate is 5%. In the base year 2025, the annual death probabilities are 0.020 at the first attained age and 0.030 at the next attained age. Mortality improves by 2% per calendar year, applied multiplicatively to each base-age probability. Compare expected present-value claims for otherwise identical policies issued at the first age in 2025 and 2026. Assume no expenses, lapses, or other decrements.

Method. Project each cohort’s rates by attained age and calendar year, then apply the two-year death-claim formula. The discount factor is $v=1/1.05$, and $v^2=1/1.1025$.

2025 cohort. In year 1, the attained-age base probability applies in 2025, so $q_1=0.020000$. In year 2, the next-age base probability has one year of improvement: $q_2=0.030000(0.98)=0.029400$. The probability of death in year 1 is 0.020000. The probability of death in year 2 is $0.98(0.0294)=0.028812$, because the policyholder must first survive year 1. Therefore,

\begin{aligned} \operatorname{EPV}_{2025} &=100{,}000\left(\frac{0.020000}{1.05} +\frac{0.028812}{1.1025}\right)\\ &=100{,}000(0.019047619+0.026133333)\\ &=\$4{,}518.10. \end{aligned}

2026 cohort. At the first attained age, one calendar year of improvement has elapsed, so $q_1=0.020000(0.98)=0.019600$. At the next age in calendar year 2027, two years have elapsed from the base year, giving $q_2=0.030000(0.98)^2=0.028812$. The probability of death in year 2 is $0.9804(0.028812)=0.0282472848$. Hence,

\begin{aligned} \operatorname{EPV}_{2026} &=100{,}000\left(\frac{0.019600}{1.05} +\frac{0.0282472848}{1.1025}\right)\\ &=100{,}000(0.018666667+0.025621120)\\ &=\$4{,}428.78. \end{aligned}

The later issue cohort has an expected present value of claims lower by $\$4{,}518.10-\$4{,}428.78=\$89.32$. The calculation checks: both the year-1 mortality and the year-2 mortality at the relevant age are lower for the later cohort, so its death-claim cost is lower. The common mistake avoided here is using the same mortality rates for both cohorts merely because their attained-age paths are the same.

How the Stress Changes Different Contract Values

The claim calculation above also explains why the sign of a mortality effect depends on the benefit. For a death benefit, a lower death probability generally reduces expected claims, all else equal. For an annuity paying while the annuitant is alive, lower mortality increases the probability that later payments are made. For either contract, the appropriate probability must be calculated for each payment date, rather than applying a single average mortality adjustment to the total value.

For a temporary life annuity with a payment $R_k$ at the end of year $k$ if the life is then alive, the survival probability to that date is the product of annual survival probabilities. Under the same annual model,

\operatorname{EPV}^{(s)}(\text{annuity}) =\sum_{k=1}^{n}v^k R_k \prod_{r=1}^{k}(1-q_r^{(s)}).

This is an end-of-year payment model: the first payment requires survival through year 1. An annuity-due payment at issue would not require survival through that first year, and its timing must be represented separately.

Worked Example: A Mortality Improvement Stress on Annuity Payments

A hypothetical two-year life annuity pays $12,000 at the end of each year if the annuitant is alive. The effective annual interest rate is 5%. The year-1 death probability is 0.020 in both scenarios. The year-2 probability is 0.030 under a no-improvement scenario and $0.030(0.98)=0.0294$ under a scenario with one year of 2% improvement. Find the present value under each scenario, assuming no other decrements.

Method. The probability of receiving the first payment is $1-0.020=0.98$. The probability of receiving the second is survival through both years. Under improvement, this is $0.98(1-0.0294)=0.951188$; without improvement, it is $0.98(1-0.030)=0.950600$.

\begin{aligned} \operatorname{EPV}_{\mathrm{improved}} &=12{,}000\left(\frac{0.98}{1.05} +\frac{0.951188}{1.1025}\right)\\ &=12{,}000(0.933333333+0.862755556)\\ &=\$21{,}553.07,\\[4pt] \operatorname{EPV}_{\mathrm{no\ improvement}} &=12{,}000\left(\frac{0.98}{1.05} +\frac{0.950600}{1.1025}\right)\\ &=12{,}000(0.933333333+0.862222222)\\ &=\$21{,}546.67. \end{aligned}

The improvement scenario increases the annuity value by $\$21{,}553.07-\$21{,}546.67=\$6.40$. The second-year survival probability is higher, so the later payment is more likely. This check confirms the expected direction for a life-contingent annuity. The common mistake avoided is treating lower mortality as a reduction in every kind of liability; for survival-contingent benefits, its effect is generally the reverse of its effect on death claims.

Reserves and Projected Profit Under Alternative Trends

A prospective reserve at time $k$, conditional on the contract being in force, is the expected present value of future outflows less the expected present value of future inflows under the assumptions used for that reserve. Mortality improvement affects the probabilities attached to future cash flows. For a single-premium term benefit with no future premiums or expenses, the reserve immediately after surviving to time $k$ is simply the present value of the remaining death benefit under the projected mortality rates.

A profit comparison also needs a clearly stated basis and timing. In the example below, expected present-value profit is defined as the initial premium less the expected present value of claims. This isolates the mortality effect; it is not a full accounting or regulatory profit projection and excludes expenses, taxes, capital costs, and other cash flows.

Worked Example: A Later Cohort’s Reserve and Expected Profit

A hypothetical two-year term contract pays $20,000 at the end of the year of death and has a single premium of $910 at issue. The effective annual interest rate is 5%; expenses and other cash flows are zero. The 2025 base-age death probabilities are 0.020 and 0.030, and the annual mortality improvement rate is 2%. Compare a policy issued in 2025 with one issued in 2026. Calculate expected present-value profit at issue and the reserve at time 1, conditional on surviving the first year. Use whole-year end-of-year death benefits.

Method. The 2025 cohort has rates $q_1=0.020000$ and $q_2=0.029400$. The 2026 cohort has $q_1=0.019600$ and $q_2=0.028812$. The latter second-year rate is the next-age base rate reduced for two calendar years. Since the premium is paid at issue, expected present-value profit is $910$ less the expected present value of claims.

For the 2025 cohort, the expected present-value claim factor was $0.045180952$ in the preceding calculation when the benefit was $100,000. Scaling to a $20,000 benefit gives claims of $20{,}000(0.045180952)=\$903.62$. Thus expected present-value profit is $\$910-\$903.62=\$6.38$.

For the 2026 cohort, the year-1 death probability is $0.0196$, and the year-2 probability of death per original policy is $0.9804(0.028812)=0.0282472848$. Therefore,

\begin{aligned} \operatorname{EPV}_{2026}(\text{claims}) &=20{,}000\left(\frac{0.0196}{1.05} +\frac{0.0282472848}{1.1025}\right)\\ &=20{,}000(0.018666667+0.025621120)\\ &=\$885.76,\\ \operatorname{EPV}_{2026}(\text{profit}) &=\$910-\$885.76=\$24.24. \end{aligned}

At time 1, conditional on survival, one year remains. The prospective reserve is $v q_2 B$. For the 2025 cohort, it is $20{,}000(0.0294)/1.05=\$560.00$. For the 2026 cohort, it is $20{,}000(0.028812)/1.05=\$548.80$. The later cohort’s reserve is lower by $\$11.20$, and its expected present-value profit is higher by $\$24.24-\$6.38=\$17.86$. These directions are consistent with the lower projected death-claim costs. The check uses the reserve conditional on having survived year 1; it does not multiply that conditional reserve by the probability of reaching time 1.

Derivation and Common Stress-Testing Traps

The cohort claim formula follows from conditional probability. To die in year $k$, the policyholder must survive years $1$ through $k-1$, then die in year $k$. Under the annual mortality model, the probability of surviving year $r$, conditional on being alive at its start, is $1-q_r^{(s)}$. Multiplying the successive conditional survival probabilities gives the probability of reaching the start of year $k$. Multiplying by the conditional death probability in year $k$ gives the probability of death during that year:

P(\text{death in year }k) =\left[\prod_{r=1}^{k-1}(1-q_r^{(s)})\right]q_k^{(s)}.

The expected present value contribution is then this probability multiplied by the year-end benefit $B_k$ and the discount factor $v^k$. Summing the mutually exclusive year-of-death contributions gives the claim formula. If the benefit is paid at another time, the discount factor changes even if the death probabilities do not.

A frequent calculation error is to interpret a 2% improvement as subtracting two percentage points from the death probability. For a base probability of 0.030, that incorrect calculation gives $0.030-0.020=0.010$. Under the proportional convention used here, the correct one-year projection is $0.030(1-0.02)=0.0294$. The incorrect value implies a 66.7% reduction, not a 2% reduction.

Another trap is to apply improvement by policy duration without checking the base calendar year. In the examples, a 2026 issue cohort has already experienced one year of improvement at issue relative to the 2025 base year. Its second duration is in 2027, two years after the base year. Starting both cohorts at the same base rates would erase this cohort timing. In a real projection, the valuation year, historical improvements already embedded in the table, and future stress period must be distinguished to avoid counting improvement twice.

State the mortality measure. A rate applied to annual death probabilities is not automatically the same as a rate applied to a central death rate or force of mortality.
Track both age and year. A cohort changes attained age and calendar year together at each duration.
Keep the comparison controlled. When measuring mortality sensitivity, retain the same interest, benefits, expenses, and other assumptions unless the scenario explicitly changes them.
Check the contract direction. Lower mortality generally reduces death claims but increases the expected duration of life-contingent annuity payments.

These projections are useful for testing whether pricing, reserves, and projected profit respond appropriately to alternative longevity trends across cohorts. A single mortality rate or average adjustment can conceal material timing differences, especially for long-duration benefits. The calculations here are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated convention that improvement is a proportional reduction in the annual death probability.

A base-year annual death probability is 0.040. It improves by 3% per year for two calendar years. Find the projected probability. (Answer: 0.037636)
A 2025 base probability at a specified age is 0.025. The annual improvement rate is 2%, and the same age is observed in 2027. Find the projected probability. (Answer: 0.024010)
A one-year death benefit of $80,000 has a base probability of 0.015. Apply 4% annual improvement for two calendar years and calculate the expected claim payment for that year. Ignore discounting and all other decrements. (Answer: $1,105.92)
A hypothetical annuity pays $5,000 at the end of year 1 if the annuitant is alive. The base probability of death is 0.020, reduced by 2% in the improvement scenario. At 4% effective annual interest, by how much does the one-payment annuity value increase under improvement? (Answer: $1.92)
A single-premium contract pays $30,000 at the end of the next year if death occurs during that year. At the reserve date the next-year base death probability is 0.040; apply one year of 2% improvement and use an effective annual interest rate of 4%. Find the reserve conditional on being alive at the reserve date. (Answer: $1,130.77)

What You'll Learn