Initial States and Starting Distributions

Why the Initial State Matters

A multiple-state model describes a person or policy by the state it occupies at a specified time. Before asking how that state might change, the model needs a starting point: which state applies at time zero? Sometimes the answer is known, such as an eligible policyholder who is active at issue. In other settings, the model represents a randomly selected member of a group whose state is not known in advance, so the starting point is described by probabilities.

These are two ways to specify the same basic information. A known initial state is a special case of a starting distribution, with probability one assigned to that state. A general starting distribution allows positive probability on more than one possible state. It describes uncertainty about the state at the beginning of the model; it does not describe any change between states.

Definition: Let the model have a finite set of possible states $S_1,S_2,\ldots,S_m$. The initial state is the state occupied at time $0$. A starting distribution is the collection of probabilities $\alpha_i=P(\text{state at time }0=S_i)$, for $i=1,\ldots,m$. It must satisfy $\alpha_i\geq 0$ for every state and $\sum_{i=1}^{m}\alpha_i=1$.

The state labels must be mutually exclusive and collectively exhaustive for the population being modelled: at time zero, an individual is in exactly one of the listed states. Keep the order of the states fixed whenever writing a distribution. For example, if the declared order is active, disabled, dead, then $(0.8,0.2,0)$ means an 80% probability of being active, a 20% probability of being disabled, and zero probability of being dead at time zero.

For a single individual, the probabilities represent uncertainty about that individual’s initial state. For a cohort, they can also describe the proportions in each state, or the probabilities for a randomly selected cohort member. In a finite cohort, multiplying the probabilities by the number of members gives expected state counts; it need not give the actual counts unless the proportions are specified as exact.

Known States and Probability Distributions

If an individual is known to start in state $S_j$, then $\alpha_j=1$ and every other starting probability is zero. For example, with states active, disabled, and dead, a policy known to be active at the model’s starting date has distribution $(1,0,0)$. This is not an assumption about the individual’s state at a later time. It states only what is known at time zero.

When the starting state is uncertain, every probability must refer to the same defined population and the same starting time. A list of probabilities that adds to less than one is incomplete: some possible state or some probability mass has been omitted. A list that adds to more than one assigns more than certainty across mutually exclusive states. Check the state definitions and arithmetic before using the distribution.

Worked Example: A Confirmed State at Issue

A hypothetical income-protection policy enters a model at issue. The states, in the declared order, are active, disabled, and dead. Eligibility checks establish that the policyholder is active at time zero. An immediate payment is $0 in the active state, $400 if disabled, and $0 if dead. Specify the starting distribution and determine the expected immediate payment. State the method and a common error to avoid.

Method. Represent the confirmed state as a probability distribution that puts probability one on that state. Then apply the payment amount for that state.

Because the policyholder is known to be active, the starting distribution is $(1,0,0)$. Its entries are nonnegative and sum to $1+0+0=1$.

The immediate payment is $0 in the active state, so the expected payment is

1(0)+0(400)+0(0)=\$0.00.

The result is checked by looking at the known state directly: an active policyholder receives no immediate payment under the stated payment rule.

Common mistake. Assigning a positive probability to disability merely because disability is a possible model state would contradict the stated confirmation of the policyholder’s state at issue. Possible later states are not additional time-zero starting states when the initial state is known.

Worked Example: Expected State Counts and an Initial Benefit

A hypothetical portfolio has 500 in-force long-term-care policies at a valuation date. A randomly selected policy is active with probability $0.76$ and receiving care benefits with probability $0.24$. These are the only two states in this model at time zero. A monthly payment of $1,250 is due immediately for each policy already receiving care; an active policy receives no immediate payment. Find the expected number of policies in each state, the expected immediate payment per policy, and the expected total immediate payment for the portfolio.

Method. Check the distribution, multiply its probabilities by the cohort size to find expected state counts, and weight each state’s immediate payment by its starting probability.

First, the starting distribution is complete because $0.76+0.24=1$. The expected active count is

500(0.76)=380\text{ policies}.

The expected count receiving care benefits is

500(0.24)=120\text{ policies}.

The expected immediate payment per policy is the probability-weighted amount:

0.76(\$0)+0.24(\$1{,}250)=\$300.00.

Therefore, the expected total immediate payment is

500(\$300.00)=\$150{,}000.00.

As a check, using the expected number receiving care gives $120(\$1{,}250)=\$150{,}000.00$, the same result. The expected active and care counts also add to $380+120=500$.

Common mistake. The immediate payment is not $1,250 for every policy. It applies only to the 24% starting in the care-benefit state. Applying it to all 500 policies would give $625,000, which ignores the state-specific payment condition.

Combining Starting Distributions

A portfolio may be made up of subgroups with different starting distributions. If a randomly selected portfolio member comes from subgroup $g$ with probability $w_g$, and subgroup $g$ has starting probability $\alpha_{g,i}$ for state $S_i$, then the portfolio starting probability for $S_i$ is the weighted average of the subgroup probabilities:

\alpha_i=\sum_g w_g\alpha_{g,i}, \qquad \sum_g w_g=1.

This is an application of the law of total probability: a member starts in $S_i$ either by belonging to one subgroup and starting in $S_i$, or by belonging to another subgroup and starting in $S_i$. The events that identify the subgroups are mutually exclusive and cover the portfolio, so their contributions add.

The combined distribution remains valid. Each $\alpha_{g,i}$ is nonnegative and each $w_g$ is nonnegative, so every weighted sum $\alpha_i$ is nonnegative. Also,

\sum_i\alpha_i =\sum_i\sum_g w_g\alpha_{g,i} =\sum_g w_g\sum_i\alpha_{g,i} =\sum_g w_g =1,

because each subgroup’s starting probabilities sum to one. This justification also shows why the subgroup weights must describe the subgroup proportions of the portfolio, rather than being chosen separately for each state.

Worked Example: Pooling Two Subgroups

A hypothetical pension administrator reviews 1,000 historical member records. The model states, in order, are active, retired, and deceased; the records retain deceased members so all three states are represented. Sixty percent of records are from subgroup X, whose starting distribution is $(0.90,0.10,0)$. The remaining 40% are from subgroup Y, with distribution $(0.75,0.20,0.05)$. At time zero, the record-level payment is $0 for an active member, $500 for a retired member, and $2,000 for a deceased member’s designated beneficiary. Find the pooled starting distribution, expected counts in each state, and total expected payment. Treat each record as equally weighted.

Method. Weight each subgroup’s state probabilities by its share of the records, combine state by state, and then apply the pooled distribution to the payment amounts.

The active probability in the pooled portfolio is

$$ 0.60(0.90)+0.40(0.75)=0.54+0.30=0.84. $$

The retired probability is

$$ 0.60(0.10)+0.40(0.20)=0.06+0.08=0.14. $$

The deceased probability is

$$ 0.60(0)+0.40(0.05)=0+0.02=0.02. $$

Thus the pooled distribution is $(0.84,0.14,0.02)$, and its entries check: $0.84+0.14+0.02=1$. For 1,000 equally weighted records, the expected counts are $1{,}000(0.84)=840$ active, $1{,}000(0.14)=140$ retired, and $1{,}000(0.02)=20$ deceased. These expected counts sum to $840+140+20=1{,}000$.

The expected payment per record is

0.84(\$0)+0.14(\$500)+0.02(\$2{,}000) =\$0+\$70+\$40 =\$110.00.

The expected total payment is $1{,}000(\$110.00)=\$110{,}000.00$. A separate count-based check gives $140(\$500)+20(\$2{,}000)=\$70{,}000+\$40{,}000=\$110{,}000$.

Common mistake. Averaging the two subgroup distributions equally would be wrong because the subgroups do not have equal sizes. The correct weights are 0.60 and 0.40, the proportions of records in each subgroup.

Expected Initial Payments: The General Rule

Suppose an immediate payment $b_i$ is made when the individual starts in state $S_i$, and no other information affects the payment. The expected payment is the probability-weighted average of the state-specific amounts:

E[B(0)]=\sum_{i=1}^{m}\alpha_i b_i.

To justify the formula, define $I_i$ to equal 1 when the individual starts in state $S_i$, and 0 otherwise. Exactly one state applies, so the payment can be written as $B(0)=\sum_i b_i I_i$. Taking expectations and using $E[I_i]=P(\text{state at time }0=S_i)=\alpha_i$ gives

E[B(0)] =E\!\left[\sum_i b_i I_i\right] =\sum_i b_i E[I_i] =\sum_i b_i\alpha_i.

For a cohort of $N$ equally weighted members, the expected total initial payment is $N\sum_i\alpha_i b_i$. This is an expectation, not a guarantee that the actual portfolio has exactly the expected number in each state or pays exactly that total. If the actual state counts are supplied, use those counts for the actual total instead.

The rule applies to payments determined by the state at time zero. It does not by itself specify later payments, state changes, or how the state probabilities evolve. Those require additional model information. Keep the starting distribution separate from any assumptions about what happens after the model begins.

Common Checks and Exam Traps

Check the total. Starting probabilities must sum to one. If they do not, resolve the missing or duplicated probability before proceeding.
Keep the state order visible. A list such as $(0.8,0.2)$ has no unambiguous meaning unless the states and their order are stated.
Do not confuse a possible state with a starting state. A state can be permitted by the model but have zero probability at time zero, as in the confirmed-active example.
Distinguish expected counts from known counts. Multiplying a cohort size by a probability gives an expected count unless exact proportions are given.
Use subgroup weights correctly. A pooled distribution uses each subgroup’s share of the population, not an unweighted average unless the subgroup shares are equal.
Match payments to states. A state-specific immediate amount is multiplied by the probability of that starting state; it is not automatically paid to every member.

Key takeaway: Specify the states and their order, assign a nonnegative time-zero probability to each state, and check that the probabilities sum to one. A known starting state is represented by probability one on that state; subgroup distributions and state-dependent initial amounts can then be combined using probability-weighted averages.

Check Your Understanding

For each problem, use the stated state order and show the probability check before relying on the distribution.

A hypothetical policy is confirmed active at the model’s starting date. The states are active, disabled, and dead. Write its starting distribution. (Answer: $(1,0,0)$.)
A cohort of 200 policies starts in either active or disabled status, with probabilities 0.65 and 0.35, respectively. What are the expected counts in each state? If the immediate payment is $900 only for a disabled policy, find the expected total immediate payment. (Answer: 130 active, 70 disabled; $63,000.00.)
A portfolio has two subgroups. Subgroup M is 40% of the portfolio and has active/care probabilities $(0.80,0.20)$; subgroup N is 60% and has probabilities $(0.50,0.50)$. Find the pooled starting distribution in the order active, care. (Answer: $(0.62,0.38)$.)
A randomly selected record starts active, disabled, or deceased with probabilities $0.70,0.20,0.10$, respectively. Immediate payments are $0, $800, and $3,000 in those states. Find the expected payment per record and for a cohort of 250 equally weighted records. (Answer: $460.00 per record; $115,000.00 total.)
A portfolio has 800 records: 25% belong to subgroup A with distribution $(0.80,0.15,0.05)$, and 75% belong to subgroup B with distribution $(0.60,0.30,0.10)$, in the order active, care, deceased. Find the pooled distribution, expected counts, and expected total immediate payment if the state payments are $0, $400, and $1,000, respectively. (Answer: distribution $(0.65,0.2625,0.0875)$; expected counts 520, 210, and 70; total payment $154,000.00.)

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