Mortality Shocks for Annuities and Longevity Exposure

Why mortality shocks matter for annuities

For an annuity, the insurer generally pays while the annuitant survives. A mortality deterioration therefore tends to reduce expected payment duration and liability, while a mortality improvement tends to increase both. This is the reverse of the usual direction for a protection contract that pays on death. The size and timing of the effect depend on when payments are due, how long the contract is deferred, and which mortality rates are stressed.

This tutorial treats a level shock: a change to mortality rates that applies consistently across the projection years specified in the scenario. It does not model a separate annual cohort trend. All examples use annual mortality probabilities and annual effective interest. Unless stated otherwise, benefits are paid at the end of a year if the annuitant is alive then, and there are no premiums or expenses after the valuation date. The examples are hypothetical and use deliberately simple assumptions to make the mechanics visible. The treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $q_k$ be the probability that a life alive at the start of year $k$ dies during that year, and let $p_k=1-q_k$. The probability of surviving $t$ full years from the valuation date is ${}_tp=\prod_{k=1}^{t}p_k$, with ${}_0p=1$. A mortality deterioration increases death probabilities; a longevity improvement decreases them. A level shock applies the specified change to each affected year, subject to the scenario’s stated rule.

Projecting annuity payments under a shock

For an annuity paying $b_t$ at the end of year $t$ only if the annuitant is then alive, the expected present value at the valuation date is the sum of each payment multiplied by its survival probability and discount factor. With annual effective interest rate $i$, $v=1/(1+i)$. For a finite annuity with payments through year $n$, the prospective value is:

V_0=\sum_{t=1}^{n} b_t\,{}_tp\,v^t

This expression applies when payments are contingent on survival to their payment dates, the benefit amounts are known, and interest and mortality assumptions are specified. It is not a formula for an annuity certain: omitting the survival factors would value payments even after death. For an annuity with annual rates that differ by year, use the product of the appropriate survival probabilities rather than raising one annual probability to a power.

Derivation: Define an indicator $I_t$ that equals 1 if the annuitant is alive at the end of year $t$, and 0 otherwise. The present value of the year-$t$ payment is $b_t v^t I_t$. Since $E[I_t]=P(I_t=1)={}_tp$, its expected present value is $b_t v^t{}_tp$. By additivity of expectation, the expected present value of all payments is $\sum_{t=1}^{n}b_t v^t{}_tp$. Independence between different payment indicators is not needed for this expectation.

A prospective reserve at a later duration uses the same logic, but conditions on the life being alive at that valuation date. It includes only future payments. For a deferred annuity, the deferment period affects the probability of reaching the first payment date; it does not make payments during that period payable.

Worked Example: Mortality Deterioration for an In-Payment Annuity

A hypothetical in-payment annuity pays $1,000 at the end of each of the next two years if the annuitant is alive. The annual mortality rate is 20% in each year on the valuation basis. A deterioration stress raises each annual rate to 30%. The effective annual interest rate is 25%, so $v=1/1.25=0.8$. Calculate the liability on each basis and the change.

Method. Find the one-year survival probability under each basis, project survival to each payment date, and discount each expected payment. This avoids treating the annuity as certain or using the death probability where a survival probability is required.

On the valuation basis, $q=0.20$, so $p=1-0.20=0.80$. Survival to year 1 is $0.80$; survival to year 2 is $0.80^2=0.64$. The present value factors, including survival, are:

\begin{aligned} \text{Year 1 factor}&=0.80(0.8)=0.64,\\ \text{Year 2 factor}&=0.64(0.8^2)=0.64(0.64)=0.4096. \end{aligned}

Thus the basis liability is $1{,}000(0.64+0.4096)=\$1{,}049.60$.

Under the stress, $q'=0.30$, so $p'=0.70$. Survival to the payment dates is $0.70$ and $0.70^2=0.49$. Therefore:

\begin{aligned} \text{Stressed year 1 factor}&=0.70(0.8)=0.56,\\ \text{Stressed year 2 factor}&=0.49(0.64)=0.3136,\\ V_0'&=1{,}000(0.56+0.3136)=\$873.60. \end{aligned}

The liability change is $\$873.60-\$1{,}049.60=-\$176.00$. The deterioration reduces the expected value of annuity payments. As a check, both stressed survival probabilities are below their basis counterparts, so with positive benefits and discount factors the stressed liability must be lower.

Deferred annuities and prospective reserves

For a deferred annuity, survival through the deferment period is part of the value at issue. If the first payment is at the end of year $m+1$, then the first payment is weighted by ${}_{m+1}p$, not merely by survival during the deferment period. At a later reserve date, however, the probability of survival before that date is already known through the conditioning that the annuitant is alive. The reserve then reflects only survival from the reserve date onward.

This distinction matters when comparing values at issue with reserves at a later duration. A shock can change the probability of reaching the first payment date and also the probabilities of receiving later payments. For an annuity already in payment, there is no remaining deferment period, but survival to each future payment still matters.

Worked Example: Longevity Improvement for a Deferred Annuity

A hypothetical deferred annuity has no payment at the end of year 1. It pays $1,000 at the ends of years 2 and 3 if the annuitant is alive. The valuation basis has a constant annual mortality rate of 20%; a longevity-improvement stress reduces it to 10% in both years and beyond. The effective annual interest rate is 25%, so $v=0.8$. Calculate the value at issue on both bases and the reserve at the end of year 1, conditional on survival to that date.

Method. At issue, include survival from issue through each payment date. At the end of year 1, condition on being alive and value the remaining payments using only future survival. This prevents counting the first year of survival twice in the reserve.

On the basis, $p=1-0.20=0.80$. At issue, survival to year 2 is $0.80^2=0.64$, and survival to year 3 is $0.80^3=0.512$. The present value factors are:

\begin{aligned} \text{Year 2 factor}&=0.64(0.8^2)=0.64(0.64)=0.4096,\\ \text{Year 3 factor}&=0.512(0.8^3)=0.512(0.512)=0.262144. \end{aligned}

The basis value at issue is $1{,}000(0.4096+0.262144)=\$671.744$, or $671.74 to the nearest cent.

Under the improvement stress, $p'=1-0.10=0.90$. Survival to years 2 and 3 is $0.90^2=0.81$ and $0.90^3=0.729$. Therefore:

\begin{aligned} \text{Stressed year 2 factor}&=0.81(0.64)=0.5184,\\ \text{Stressed year 3 factor}&=0.729(0.512)=0.373248,\\ V_0'&=1{,}000(0.5184+0.373248)=\$891.648. \end{aligned}

The issue value increases by $\$891.648-\$671.744=\$219.904$, or $219.90 to the nearest cent.

At the end of year 1, conditional on being alive, the basis reserve covers the year-2 and year-3 payments. The remaining survival probabilities are $0.80$ and $0.80^2=0.64$, giving:

V_1=1{,}000\left[0.80(0.8)+0.64(0.8^2)\right] =1{,}000(0.64+0.4096)=\$1{,}049.60.

Under the stress, the corresponding conditional survival probabilities are $0.90$ and $0.90^2=0.81$:

V_1'=1{,}000\left[0.90(0.8)+0.81(0.8^2)\right] =1{,}000(0.72+0.5184)=\$1{,}238.40.

The reserve at year 1 increases by $\$188.80$. The increase is smaller than the issue-value increase because the issue value also reflects the stressed chance of surviving the first year. The reserve is conditional on survival to its valuation date.

Mortality results and asset shares

A mortality shock changes projected claims as well as liabilities. For an annuity book, fewer deaths than assumed generally mean more benefit payments than allowed for by the basis; this is a mortality loss relative to that basis. More deaths generally mean fewer annuity payments and a mortality gain. The sign is from the insurer’s perspective and assumes benefit amounts, investment returns, expenses, and all other assumptions are held fixed.

An asset share tracks funds allocated to a policy or group as premiums and investment income accumulate, less expenses and benefits paid. In a simplified projection with starting allocated assets $A_0$, an immediate expense $E$, annual effective investment return $i$, and end-of-year claims $C$, the amount remaining after claims is $(A_0-E)(1+i)-C$. If the only changed assumption is mortality, the difference in projected claims directly changes the amount remaining. A fuller projection may include future premiums, expenses, taxes, or other cash flows; they must be included consistently on both scenarios.

Worked Example: Mortality Gain and Projected Asset Share

A hypothetical group of 100 in-payment annuitants has $100,000 of allocated assets at the start of a year. An expense of $5,000 is paid immediately. The assets earn an effective annual return of 5%, and each annuitant alive at year-end receives $1,000. The basis mortality rate is 20%; a deterioration scenario uses 30%. For a simple projected comparison, use the expected survivor counts, 80 on the basis and 70 under stress. Calculate year-end assets after claims, the mortality result relative to basis, and the residual asset share per projected survivor.

Method. Deduct the immediate expense, apply investment return to the balance, then subtract expected claims. Compare projected claims with the basis before interpreting the mortality result. This order avoids earning a full year of interest on an expense paid at the start.

Assets immediately after expense are $\$100{,}000-\$5{,}000=\$95{,}000$. Just before year-end claims, they are $\$95{,}000(1.05)=\$99{,}750$.

On the basis, expected survivors are $100(1-0.20)=80$. Expected claims are $80(\$1{,}000)=\$80{,}000$, leaving $\$99{,}750-\$80{,}000=\$19{,}750$. The residual asset share per projected survivor is $\$19{,}750/80=\$246.875$, or $246.88 to the nearest cent.

Under the deterioration stress, expected survivors are $100(1-0.30)=70$. Expected claims are $70(\$1{,}000)=\$70{,}000$, leaving $\$99{,}750-\$70{,}000=\$29{,}750$. The residual asset share per projected survivor is $\$29{,}750/70=\$425.00$.

Claims are $\$10{,}000$ lower under stress, so the projected mortality result relative to basis is a $10,000 gain for the insurer in this one-year illustration. The remaining assets are divided among fewer projected survivors, increasing the per-survivor residual asset share. The calculation is a simplified projection, not a claim that actual group survivor counts must equal their expected values.

Interpreting the direction and avoiding common traps

For a fixed positive annuity benefit and fixed interest assumptions, reducing mortality increases survival to each future payment date. Each term in the expected present value is therefore at least as large as before, so the liability cannot decrease. Increasing mortality has the opposite effect. This directional check is useful, but it does not replace calculation: the amount of the change depends on the payment schedule, deferment, duration, and discounting.

A common mistake is to say that a mortality deterioration increases the annuity liability because “risk has increased.” Here is the wrong direction for the first worked example: it treats the increase from 20% to 30% mortality as if survival rose from 80% to 90%, producing a year-1 factor of $0.90(0.8)=0.72$, greater than the basis factor $0.64$. The correction is $p'=1-q'=1-0.30=0.70$, so the correct year-1 factor is $0.70(0.8)=0.56$. The stressed liability is consequently lower, not higher.

Another trap is to apply a shock on a different scale from the one specified. A 10% relative increase in $q_k$ means $q_k'=1.10q_k$, while a 10 percentage-point increase means $q_k'=q_k+0.10$. A shock to the force of mortality $\mu_x$ is not automatically the same as either change to an annual death probability. Convert or apply the shock according to the problem’s stated mortality model; do not silently substitute one interpretation for another. Also check that the resulting probabilities remain valid, between 0 and 1.

Finally, keep valuation dates and cash-flow timing explicit. A payment at the end of year $t$ is discounted by $v^t$, and the relevant survival probability is survival to that payment date. A reserve at time $t$, conditional on survival to time $t$, starts with future survival from that date—not survival from issue all over again.

Key takeaway: For annuities, lower mortality generally increases expected benefits, prospective reserves, and the strain on assets; higher mortality generally reduces them. Project survival to each payment date, condition correctly at reserve dates, and compare asset shares on a consistent basis.

Check Your Understanding

Use annual effective interest and end-of-year payments unless a question states otherwise. Round money to the nearest cent.

A one-year in-payment annuity pays $2,000 if the annuitant survives the year. Mortality falls from 15% to 12%, and $i=4\%$. By how much does its expected present value increase? (Answer: $57.69)
A two-year in-payment annuity pays $500 at each year-end if alive. The annual mortality rate rises from 10% to 20%, with $i=0\%$. Find the change in value, stressed less basis. (Answer: −$135.00)
A deferred annuity pays $1,000 at the end of year 2 if alive, with no other payments. Mortality falls from 10% to 5% in both years, and $i=5\%$. Find the increase in value at issue. (Answer: $83.90)
A group has 50 annuitants and $40,000 in allocated assets. An immediate expense of $1,000 is followed by investment growth at 5%; the year-end payment is $500 per survivor. Compare basis mortality of 20% with stressed mortality of 10%. Give the remaining assets under each basis and the per-survivor residual asset shares. (Answer: Basis: $20,950 remaining and $523.75 per projected survivor; stress: $18,450 remaining and $410.00 per projected survivor.)
A deferred annuity pays $1,000 at the ends of years 2 and 3 if alive. At the end of year 1, conditional on survival, calculate its reserve on mortality of 20% and on improved mortality of 10%, with $i=25\%$ in both cases. (Answer: Basis reserve $1,049.60; improved-mortality reserve $1,238.40; increase $188.80.)

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