Mortality Gains and Losses

Why Mortality Experience Changes Surplus

For a life insurer, deaths affect both benefit payments and the obligations that remain for survivors. If actual deaths exceed expected deaths, surplus does not automatically decrease: the outcome depends on how the cost of a death compares with the cost of a survivor. A large death claim may make additional deaths costly, while a large survival benefit or reserve may mean that additional deaths reduce costs.

This tutorial isolates the mortality component of surplus for a group of policies over one period. The comparison holds other experience, such as investment returns and expenses, fixed. All amounts in the examples are hypothetical. They are measured at the end of the period, so no discounting is needed within the one-period calculations.

Definition: For a group of $n$ policies with expected death probability $q$, expected deaths are $nq$. Let $C_D$ be the insurer’s end-of-period cost for a policy ending in death, and $C_S$ its end-of-period cost for a policy surviving the period. The mortality gain or loss, relative to expected experience, is $$ G_{\mathrm{mort}}=(nq-D)(C_D-C_S), $$ where $D$ is the actual number of deaths. A positive result is a gain in surplus; a negative result is a loss.

The cost for each status includes the payment made at the end of the period and, where applicable, the liability remaining immediately after that payment. Thus, for a policy with a death benefit $B_D$ and no continuing liability after death, $C_D=B_D$. For a surviving policy with a survival benefit $B_S$ and a closing reserve $V_1$ after that benefit is paid, $C_S=B_S+V_1$. If there is no survival payment or closing reserve, the corresponding amount is zero.

The formula is a comparison with expected mortality, not a full profit test. Premiums and other cash flows common to the two outcomes cancel in the comparison. If a cash flow differs by death or survival status, it belongs in the appropriate status cost. The formula does not by itself allow for investment, expense, lapse, or other experience differences.

Deriving the Mortality Gain Formula

Under the stated model, assume a homogeneous group: each policy has the same expected death probability $q$, and its cost depends only on whether it ends the period in death or survival. The actual total end-of-period cost for $D$ deaths is $D C_D+(n-D)C_S$. At expected mortality, the corresponding expected cost is $nqC_D+n(1-q)C_S$. The mortality gain is expected cost minus actual cost, because a lower actual cost than expected increases surplus.

\begin{aligned} G_{\mathrm{mort}} &=\left[nqC_D+n(1-q)C_S\right] -\left[DC_D+(n-D)C_S\right]\\ &=(nq-D)C_D+\left[n(1-q)-(n-D)\right]C_S\\ &=(nq-D)C_D+(D-nq)C_S\\ &=(nq-D)(C_D-C_S). \end{aligned}

This derivation shows why the sign depends on the difference between the two status costs. When $C_D>C_S$, a death costs more than a survivor, so actual deaths above expectation reduce surplus. When $C_D<C_S$, the direction reverses: more deaths than expected can increase surplus.

For a homogeneous group, the assumed number of deaths can be modeled as $D\sim\operatorname{Binomial}(n,q)$ when lives die independently with the same probability $q$. The formula for the realized gain uses the observed count $D$ and the expected count $nq$; it does not require calculating the probability of that observed count. If policies have different mortality assumptions or status costs, calculate expected and actual costs policy by policy, or within suitably homogeneous groups, rather than applying one group rate and one cost difference indiscriminately.

Death Benefits: Excess Deaths Can Create a Loss

Worked Example: Term Death Coverage With More Deaths Than Expected

A hypothetical one-year term portfolio has 500 identical policies. The expected annual death probability is $q=0.01$; assume independent deaths, so $D\sim\operatorname{Binomial}(500,0.01)$ under the mortality assumption. Actual deaths are 7. Each death benefit is $100,000, and there is no survival benefit or closing reserve. Use an effective annual interest rate of $i=0\%$ for the one-period comparison because all costs are measured at the end of the year. Find the mortality gain or loss.

Step 1: Find expected deaths. The expected count is $nq=500(0.01)=5$. Thus actual deaths exceed expected deaths by $7-5=2$.

Step 2: Determine the status costs. A death costs $C_D=\$100{,}000$. Survival costs $C_S=\$0$, since there is no survival payment or remaining reserve.

Step 3: Apply the mortality formula.

G_{\mathrm{mort}}=(5-7)(\$100{,}000-\$0)=-\$200{,}000.

The result is a $200,000 mortality loss. As a check, the expected claim cost is $5(\$100{,}000)=\$500{,}000$, while actual claim cost is $7(\$100{,}000)=\$700{,}000$. Actual cost is $200,000 higher. The method compares expected and actual claim costs; it avoids treating the excess two deaths as a gain merely because the count difference is positive.

Survival Benefits: More Deaths May Reduce Costs

When a benefit is payable only if the policyholder survives to the end of the period, a death can avoid that payment. In this case, deaths above expectation may create a gain. This is not a contradiction: the formula compares the costs of the two outcomes, rather than assigning a fixed favorable or unfavorable meaning to mortality.

Worked Example: Survival Benefit With Excess Deaths

A hypothetical group has 200 identical policies, each paying a $20,000 survival benefit at the end of one year and no death benefit. The expected annual death probability is $q=0.02$; assume independent deaths, with $D\sim\operatorname{Binomial}(200,0.02)$. There is no closing reserve after the benefit is paid. Actual deaths are 6. Use $i=0\%$ for this end-of-year comparison. Calculate the mortality result.

Step 1: Calculate expected deaths and expected survivors. Expected deaths are $200(0.02)=4$, so expected survivors are $200-4=196$. Actual survivors are $200-6=194$.

Step 2: Set the costs for the two outcomes. A death has no payment or continuing liability, so $C_D=\$0$. A survivor receives $20,000 and has no remaining reserve, so $C_S=\$20{,}000$.

Step 3: Calculate the gain.

G_{\mathrm{mort}}=(4-6)(\$0-\$20{,}000)=\$40{,}000.

There is a $40,000 mortality gain. Check by comparing costs: expected survival-benefit cost is $196(\$20{,}000)=\$3{,}920{,}000$, while actual cost is $194(\$20{,}000)=\$3{,}880{,}000$. The actual cost is $40,000 lower. This conclusion depends on the stated benefit design; it is not a general claim that higher mortality is favorable.

Mixed Benefits and Closing Reserves

Some contracts pay different benefits depending on whether the policyholder dies or survives, and surviving policies may carry a reserve into the next period. For these contracts, comparing the death benefit alone with the survival benefit alone can give the wrong answer. The survivor’s closing reserve is also an insurer liability, so it is included in $C_S$.

Worked Example: Mixed End-of-Year Benefits

A hypothetical portfolio has 300 identical policies. The expected one-year death probability is $q=0.02$; assume independent deaths, with $D\sim\operatorname{Binomial}(300,0.02)$. A death at year-end produces a $42,000 payment. A survivor receives $7,000 at year-end, after which the contract ends with no closing reserve. Actual deaths are 9. Use an effective annual interest rate of $i=0\%$, since all amounts are measured at the same year-end. Find the mortality gain or loss.

Step 1: Find expected deaths. Expected deaths are $300(0.02)=6$, so there are $9-6=3$ excess deaths.

Step 2: Compare the costs by status. The death-path cost is $C_D=\$42{,}000$. The survivor-path cost is $C_S=\$7{,}000+\$0=\$7{,}000$. The difference is $\$42{,}000-\$7{,}000=\$35{,}000$.

Step 3: Calculate the mortality result.

G_{\mathrm{mort}}=(6-9)(\$35{,}000)=-\$105{,}000.

The portfolio has a $105,000 mortality loss. Check using total costs: expected cost is $6(\$42{,}000)+294(\$7{,}000)=\$252{,}000+\$2{,}058{,}000=\$2{,}310{,}000$. Actual cost is $9(\$42{,}000)+291(\$7{,}000)=\$378{,}000+\$2{,}037{,}000=\$2{,}415{,}000$. Actual cost exceeds expected cost by $105,000.

Reserves, Sign Checks, and Common Traps

For an in-force contract, a survivor’s closing reserve can materially change the mortality result. For example, if $C_D=\$50{,}000$, the survivor receives $10,000, and the closing reserve after that payment is $15,000, then $C_S=\$10{,}000+\$15{,}000=\$25{,}000$. An excess death costs an additional $25,000 relative to survival, not $40,000 relative to the survival payment alone. The reserve is included as a liability; it is not treated as a cash payment.

Set the comparison basis.
Identify the policy group, period, expected mortality probability, and observed death count. Compute expected deaths as $nq$ for a homogeneous group.

Build each status cost.
Include the payment at the end of the period and any liability remaining after it. Keep common cash flows out of the difference because they cancel.

Apply and interpret the sign.
Calculate $(nq-D)(C_D-C_S)$. A positive value is a gain; a negative value is a loss.

Check by total costs.
Compare expected and actual costs. Expected cost minus actual cost must equal the stated mortality gain or loss.

A frequent exam trap is reversing the death-count difference. In the mixed-benefit example, the incorrect calculation $(9-6)(\$42{,}000-\$7{,}000)=+\$105{,}000$ labels excess deaths a gain. That sign is wrong: it multiplies excess deaths by the extra cost per death, which increases actual cost. The correct surplus comparison is expected deaths minus actual deaths: $(6-9)(\$35{,}000)=-\$105{,}000$, a loss. A second trap is ignoring a survivor’s closing reserve, which understates $C_S$ and overstates the cost difference between death and survival.

Mortality gains and losses are one component of participating-policy surplus. The calculation establishes how mortality experience affects surplus before any later decision about its allocation. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus, where precise treatment of experience assumptions, benefit timing, and reserves is essential.

Key takeaway: Mortality experience affects surplus according to both the deviation of actual deaths from expected deaths and the cost difference between death and survival. Include benefits and the survivor’s closing reserve, then verify the sign by comparing total expected and actual costs.

Check Your Understanding

For each hypothetical one-period problem, treat all amounts as measured at the end of the period, use $i=0\%$, and hold non-mortality experience fixed. State whether the result is a gain or loss and its amount.

A group has 240 policies, expected death probability $q=0.025$, and 4 actual deaths. Death costs $80,000; a survivor receives $5,000 and has no closing reserve. Assume independent deaths under the mortality model. (Answer: $150,000 gain)
A group has 600 policies, expected death probability $q=0.012$, and 10 actual deaths. Death costs $30,000; survival costs $2,000, with no closing reserve. Assume independent deaths under the mortality model. (Answer: $78,400 loss)
A group has 150 policies, expected death probability $q=0.04$, and 5 actual deaths. Death costs $50,000; a survivor receives $10,000 and has a $15,000 closing reserve after payment. Assume independent deaths under the mortality model. (Answer: $25,000 gain)
A group has 400 policies, expected death probability $q=0.015$, and 8 actual deaths. Death costs $70,000; survival costs $10,000, with no closing reserve. Assume independent deaths under the mortality model. (Answer: $120,000 loss)
A group has 250 policies, expected death probability $q=0.02$, and 7 actual deaths. Death costs $36,000; a survivor receives $12,000 and has a $8,000 closing reserve. Assume independent deaths under the mortality model. (Answer: $32,000 loss)

What You'll Learn