Combined Mortality and Morbidity Scenarios

Why Combine Mortality and Morbidity?

A long-term policy can be affected by mortality and morbidity at the same time. For example, higher disability incidence can increase disability claims, while a change in mortality can alter both death claims and the number of people who remain eligible for future disability benefits. Looking at each assumption separately may therefore miss the combined effect on the policy’s cash flows and liabilities.

A combined scenario specifies a set of transition assumptions that apply together. It is a defined stress, not a claim that the assumption changes are statistically independent or that they occur with a particular probability. The essential modelling requirement is that the transitions remain coherent: conditional on the current state, their probabilities must be nonnegative and sum to one.

Definition: A combined mortality and morbidity scenario is a specified set of transition probabilities or intensities in which mortality and morbidity assumptions are changed together. In a discrete-time model, let $p_{sj}^{(k)}$ be the probability that a policy moves from state $s$ to state $j$ during a period under scenario $k$. For every starting state $s$, the probabilities must satisfy $p_{sj}^{(k)} \geq 0$ and $\sum_j p_{sj}^{(k)}=1$. The scenario must also specify the benefits and other cash flows associated with those transitions or destination states.

For clarity, consider three states: healthy ($H$), disabled ($D$), and dead ($X$). Death is absorbing, so a policy already in $X$ remains there. Starting from $H$, possible transitions over one year might be $H$ to $D$, $H$ to $X$, or $H$ to $H$. These outcomes compete: a person cannot make both the healthy-to-disabled and healthy-to-dead transition during the same annual step.

p_{HH}+p_{HD}+p_{HX}=1

The same check applies to the disabled state. A policy in $D$ may recover to $H$, die, or remain disabled. Raising disability incidence while also raising mortality does not mean multiplying the original transition probabilities by two stress factors and leaving the remaining probability unchanged. Recalculate the residual probability after specifying the stressed exits.

From a Joint Scenario to Expected Claims

Let $b_{sj}$ be the benefit paid at the end of the period when the policy moves from state $s$ to state $j$. The expected benefit for a policy beginning in state $s$ is the sum of each transition probability multiplied by its associated payment. If a benefit depends on the state at year-end rather than on the transition itself, define $b_{sj}$ accordingly. State explicitly whether a death benefit and a disability benefit can both be paid in the same period; in the examples below, a death transition pays the death benefit and a disabled destination pays the disability benefit.

E[B\mid s,k]=\sum_j p_{sj}^{(k)}b_{sj}

This conditional expectation provides the period claim cost for a policy in state $s$, under scenario $k$. If there are policies in several starting states, multiply each state’s expected payment by the number of policies in that state and add the results. A change to several transitions can affect claims in different directions; the calculation, rather than a general rule about “higher mortality,” determines the net effect.

Worked Example: Combined Stress on One-Year Claims

A hypothetical one-year contract starts with one healthy insured. At year-end, it pays $24,000 if the insured is disabled and $5,000 if the insured dies. The outcomes are mutually exclusive, and there are no other benefits or expenses. Use an annual effective interest rate of $i=0\%$, so no discounting is needed. Under the base assumptions, healthy-to-disabled incidence is 0.08 and healthy-to-death probability is 0.01. Under the combined stress, incidence is 0.12 and healthy-to-death probability is 0.02. Calculate the expected claim under each set of assumptions and check the stressed transition probabilities.

Method. Multiply each mutually exclusive outcome probability by its payment, then add. The probability of remaining healthy is the residual probability.

Under the base assumptions, the probability of remaining healthy is $1-0.08-0.01=0.91$. The transition probabilities sum to $0.08+0.01+0.91=1.00$. Expected claims are:

0.08(24{,}000)+0.01(5{,}000)+0.91(0) =1{,}920+50 =\$1{,}970

Under the combined stress, the probability of remaining healthy is $1-0.12-0.02=0.86$, and the transition probabilities sum to $0.12+0.02+0.86=1.00$. Thus:

0.12(24{,}000)+0.02(5{,}000)+0.86(0) =2{,}880+100 =\$2{,}980

The expected claim increases by $\$2{,}980-\$1{,}970=\$1{,}010$. The increase includes $\$960$ from the additional disability probability and $\$50$ from the additional death probability. Check: stressed probabilities are nonnegative and sum to one, and the expected claim is the sum of the two mutually exclusive benefit amounts.

Common mistake avoided: Do not treat disabled and dead as simultaneous outcomes and add their probabilities as though both benefits can be paid on one policy in one year. Here the two transition probabilities identify competing outcomes; each payment is weighted by its own probability.

Prospective Reserves Under a Joint Scenario

A multi-state prospective reserve is conditional on the policy’s current state. In a one-year discrete model, let $V_j$ be the reserve at the end of the year if the policy survives in state $j$, and set the reserve in the dead state to zero. Suppose the year-end benefit for a transition from $s$ to $j$ is $b_{sj}$, the annual effective interest rate is $i$, and there are no premiums or expenses in this reserve calculation. Conditioning on the next state gives the reserve recursion below.

V_s=\frac{\sum_j p_{sj}\left(b_{sj}+V_j\right)}{1+i}

The term $b_{sj}$ is paid during the year, and $V_j$ is the liability remaining at year-end if the policy reaches state $j$. Both are included before discounting. In a more detailed model, premiums and expenses must be placed at their actual payment times and included with their appropriate signs. The equation does not apply unchanged if the timing convention or cash flows differ.

Justification: Conditional on transition $s$ to $j$, the year-end amount represented by the current benefit and continuing reserve is $b_{sj}+V_j$. Its expected value, conditional on starting in $s$, is $\sum_j p_{sj}(b_{sj}+V_j)$. Discounting that expected year-end liability for one year at effective annual rate $i$ gives the value at the start of the year, hence division by $1+i$.

For a combined scenario, use the joint scenario’s transition probabilities in the recursion. A change in current-year transition probabilities affects the weights on both the current benefit and the end-of-year state reserve. This is why a reserve impact cannot generally be calculated from the change in current-year claims alone.

Worked Example: Revaluing a Healthy-State Reserve

A hypothetical policy starts healthy. During the year, a disabled destination pays $1,000 and a death transition pays $4,000; a healthy destination pays nothing. The annual effective interest rate is $i=4\%$. At year-end, the continuing reserves are $300 in state $H$, $900 in state $D$, and $0 in state $X$. Treat those state-specific reserves as fixed continuation liabilities for this calculation. Base transitions from $H$ are $p_{HD}=0.10$ and $p_{HX}=0.02$. The combined stress changes them to $p_{HD}=0.15$ and $p_{HX}=0.03$. Calculate the base and stressed starting reserves.

Method. For each basis, find the residual probability of remaining healthy, calculate the expected year-end claim plus continuing reserve, and discount by $1.04$.

On the base basis, $p_{HH}=1-0.10-0.02=0.88$. The expected current benefit is $0.10(\$1{,}000)+0.02(\$4{,}000)=\$180$. The expected continuing reserve is $0.88(\$300)+0.10(\$900)+0.02(\$0)=\$354$. Therefore:

V_H^{\text{base}}=\frac{180+354}{1.04} =\frac{534}{1.04} =\$513.4615\ldots

Under stress, $p_{HH}=1-0.15-0.03=0.82$. The expected current benefit is $0.15(\$1{,}000)+0.03(\$4{,}000)=\$270$. The expected continuing reserve is $0.82(\$300)+0.15(\$900)+0.03(\$0)=\$381$. Thus:

V_H^{\text{stress}}=\frac{270+381}{1.04} =\frac{651}{1.04} =\$625.9615\ldots

Rounded to the nearest cent, the base reserve is $513.46 and the stressed reserve is $625.96, an increase of $\$625.96-\$513.46=\$112.50$. Check: before rounding, the increase is $(651-534)/1.04=117/1.04=\$112.50$.

Common mistake avoided: Do not discount only the expected current-year benefits and omit the continuing liability. The policy can remain alive in $H$ or $D$, so the end-of-year reserves in those states contribute to the starting reserve.

Profit: Keep the Opening and Closing Bases Clear

A profit calculation must state its timing and reserve convention. The following one-year convention treats the opening reserve as an asset available at the start of the year. A premium is also received at the start of the year, and both accumulate at rate $i$. Claims, expenses, and the closing reserve are measured at year-end. For expected profit, use expected claims and the expected closing reserve under the specified transitions.

\text{Expected year-end profit} =(V_0+P)(1+i)-E[B]-E[E]-E[V_1]

Here $V_0$ is the opening reserve, $P$ is the start-of-year premium, $E[B]$ is expected year-end benefit, $E[E]$ is expected year-end expense, and $E[V_1]$ is the expected closing reserve. This is an asset-minus-liability calculation under the stated timing convention; it is not a formula for every profit-testing model. If premium timing, expense timing, or reserve definitions change, the equation must be adapted.

Worked Example: One-Year Profit Under a Combined Stress

Use the policy and transition assumptions in the reserve example: $i=4\%$, a healthy starting state, base probabilities $p_{HD}=0.10$, $p_{HX}=0.02$, and stressed probabilities $p_{HD}=0.15$, $p_{HX}=0.03$. The year-end benefits are $1,000 on transition to disability and $4,000 on transition to death. The year-end reserves are $300 in $H$, $900 in $D$, and $0 in $X$. A premium of $150 is received at the start of the year, and a fixed expense of $25 is paid at year-end. Set the opening reserve equal to the base reserve from the preceding calculation, before rounding. Calculate base-basis and stress-basis expected profit, keeping the same opening reserve and premium for both.

Method. Accumulate the opening reserve and premium to year-end. Then subtract each basis’s expected benefit, the $25 expense, and its expected closing reserve. Keeping the opening reserve fixed means the stress result measures profit emergence from a base opening position; it does not include a separate opening-reserve revaluation.

The opening reserve is $V_0=534/1.04$, so its accumulated value is $V_0(1.04)=\$534$. The accumulated start-of-year premium is $\$150(1.04)=\$156$. Total year-end assets are therefore $\$534+\$156=\$690$.

On the base basis, expected benefits are $180 and the expected closing reserve is $354, as calculated above. Expected profit is:

690-180-25-354=\$131

Under stress, expected benefits are $270 and the expected closing reserve is $381. Expected profit is:

690-270-25-381=\$14

The combined stress reduces expected one-year profit by $\$131-\$14=\$117$. Check: the base expected year-end claim plus reserve is $\$180+\$354=\$534$, equal to the accumulated opening reserve; the base profit is therefore the accumulated premium less the expense, or $\$156-\$25=\$131$. The stress adds $\$90$ of expected benefits and $\$27$ of expected closing reserve, reducing profit by $\$117$.

Common mistake avoided: Do not subtract the stressed opening reserve as well as the stressed closing reserve in this calculation. The stated convention holds the opening reserve fixed at its base value. A separate opening reserve revaluation is a different measurement and must be identified rather than silently mixed into one-year profit.

Practical Checks for Combined Scenarios

Before interpreting a result, check the scenario construction and the cash-flow timing. These checks are especially important when mortality and morbidity assumptions change together, because an invalid transition row can create implausible claim and reserve results even if the arithmetic is otherwise correct.

Specify the starting state and time step.
State whether probabilities apply over a month, a year, or another interval, and identify the state occupied at the start of the period.

Set the joint transitions.
List the stressed mortality and morbidity transitions together. Calculate any residual stay probability and verify that every row sums to one.

Map transitions to payments.
Identify which cash flow is paid on each transition or destination state, and confirm that mutually exclusive outcomes are not counted as simultaneous claims.

Value current and continuing liabilities.
Use the scenario probabilities for current benefits and for the states that determine closing reserves. Apply discounting only after aligning the cash flows to the valuation date.

State the profit convention.
Identify whether the opening reserve is fixed or revalued, and place premiums, expenses, claims, and closing reserves at their stated payment times.

A combined stress is useful because a policy’s exposure is defined by its full transition structure, not by one assumption in isolation. A mortality change may increase death claims but reduce the probability of future disability payments; a morbidity change may increase the number entering a disabled state and alter continuing liabilities. The direction and size of the net result depend on the scenario, benefit design, starting state, and timing. The earlier material on mortality and disability stresses provides the individual assumption context; the combined calculation must still be performed using a coherent joint set of transitions.

Key takeaway: Build the joint transition model first, verify its probabilities, and then calculate claims, reserves, and profit using the same scenario and explicit timing conventions. Never infer a combined result by simply adding isolated stress results unless the cash-flow structure justifies that calculation.

Check Your Understanding

For each problem, state or use the indicated timing and probability conventions. Answers are provided for checking.

A one-year policy starts healthy. It pays $8,000 on a disability transition and $2,000 on a death transition. Under a combined scenario, the probabilities are 0.09 for disability and 0.015 for death; all other outcomes pay nothing. At $i=0\%$, what is the expected claim? (Answer: $750.)
For a healthy starting state, a stressed annual transition probability to disability is 0.14 and to death is 0.025. What is the probability of remaining healthy, and do the three outcomes form a valid transition row? (Answer: 0.835; yes, the probabilities sum to 1.000.)
A policy starts healthy. At year-end it pays $500 on transition to disability and $2,000 on transition to death. The annual effective rate is 5%. The base probabilities are 0.10 to disability and 0.02 to death; the year-end reserves are $100 in $H$, $400 in $D$, and $0 in $X$. Calculate the base prospective reserve. (Answer: $(50+40+88+40)/1.05=\$207.62$.)
A disabled policy has annual transition probabilities 0.30 to recovery, 0.10 to death, and 0.60 to remaining disabled. At year-end, the reserves in recovered, dead, and still-disabled states are $250, $0, and $700, respectively. What is the expected year-end reserve? (Answer: $0.30(250)+0.10(0)+0.60(700)=\$495$.)
Under a one-year profit convention, the opening reserve is $300, a $80 premium is received at the start of the year, and $i=4\%$. Expected year-end claims are $120, expenses are $10, and the expected closing reserve is $250. Calculate expected profit. (Answer: $(300+80)(1.04)-120-10-250=\$15.20$.)

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