State-Dependent Premiums and Expenses

Why State-Specific Cash Flows Matter

In a multi-state contract, the cash flows can change when the insured changes state. Premiums may be payable only while a person is active, while expenses may continue during disability or care. A Thiele equation must therefore use the cash-flow rates for the state the policy occupies at time $t$, not a single premium or expense rate applied to every state.

The reserve $V_j(t)$ is the prospective reserve at time $t$, conditional on the contract being in state $j$. Its derivative describes how that reserve changes locally while the contract remains in that state. The transition terms account for the possibility of leaving state $j$; state-specific continuous rates account for premiums, expenses, and benefits paid while in $j$.

Definition: Let $V_j(t)$ be the reserve in state $j$, and let $\mu_{jk}(t)$ be the transition intensity from state $j$ to a distinct state $k$. Define $p_j(t)$ as the continuous premium income rate in state $j$, $e_j(t)$ as the continuous expense outgo rate, and $c_j(t)$ as the continuous benefit outgo rate. Let $b_{jk}(t)$ be a lump-sum benefit paid on a transition from $j$ to $k$. With force of interest $\delta(t)$, the multi-state Thiele equation is $$ V_j'(t)=\delta(t)V_j(t)+p_j(t)-e_j(t)-c_j(t)+\sum_{k\ne j}\mu_{jk}(t)\left[V_j(t)-V_k(t)-b_{jk}(t)\right]. $$ All rates are measured per unit of time, and money amounts such as $V_j$ and $b_{jk}$ are in consistent currency units. A premium that is not payable in state $j$ has $p_j(t)=0$; this does not imply that expenses or benefits in that state are also zero.

Reading Each Term in the Equation

The term $\delta V_j$ is the interest earned on the reserve. Premium income increases the reserve, so $p_j$ enters positively. Expenses and continuous benefits are outgoes, so $e_j$ and $c_j$ enter negatively. The transition contribution compares the reserve and transition payment if the policy leaves $j$ with the reserve associated with remaining in $j$. If a transition benefit is paid, it reduces the amount available after the transition.

The indices matter. The rates $p_j,e_j,c_j$ describe cash flows in the current state $j$. By contrast, the transition term uses the destination reserve $V_k$, because a transition to $k$ puts the contract into that state. For several possible destinations, include one term for each destination with a nonzero transition intensity.

Identify the current state.
Use that state's reserve and its own continuous premium, expense, and benefit rates.

List possible transitions.
For every destination $k$, record $\mu_{jk}$, $V_k$, and any transition benefit $b_{jk}$.

Substitute and check signs.
Add interest and premium income, subtract continuous outgo, and add each transition term $\mu_{jk}(V_j-V_k-b_{jk})$.

Why the Transition Term Has This Form

A short-interval reserve balance justifies the equation. Assume the forces and payment rates are locally constant over an interval of length $h$, with $h$ small. Conditional on being in state $j$ at time $t$, the probability of a transition to $k$ during the interval is $\mu_{jk}(t)h+o(h)$, and the probability of no transition is $1-\sum_{k\ne j}\mu_{jk}(t)h+o(h)$. The probability of two or more transitions is $o(h)$.

Write the continuous net outgo rate in state $j$ as $e_j+c_j-p_j$. Over a short interval, the reserve at $t$, accumulated with interest to $t+h$, must cover the expected net outgo and the reserve or benefit due at the end of the interval. To first order in $h$, the balance is

(1+\delta h)V_j(t) = \left(1-\sum_{k\ne j}\mu_{jk}h\right) \left[V_j(t)+V_j'(t)h+(e_j+c_j-p_j)h\right] +\sum_{k\ne j}\mu_{jk}h\left[V_k(t)+b_{jk}\right]+o(h).

Subtract $V_j(t)$, divide by $h$, and let $h$ decrease to zero. The terms involving the transition probabilities combine as $\sum_{k\ne j}\mu_{jk}(V_k+b_{jk}-V_j)$ on the right side of the balance. Rearranging gives

V_j'=\delta V_j-(e_j+c_j-p_j)+\sum_{k\ne j}\mu_{jk}(V_j-V_k-b_{jk}),

which is the stated Thiele equation. This balance assumes that the listed continuous flows accrue only while the contract occupies the relevant state, and that $b_{jk}$ is paid on the specified transition. A different timing or payment convention must be represented by changing the cash-flow model, not by changing signs informally.

Worked Applications

Worked Example: Premiums and Expenses Differ Across Two States

Consider a hypothetical policy with states $A$ and $B$. The force of interest is $\delta=0.04$ per year. At the time considered, $V_A=\$12{,}000$ and $V_B=\$9{,}000$. In state $A$, the premium rate is $\$1{,}200$ per year, the expense rate is $\$180$ per year, and the continuous benefit rate is $\$300$ per year. The only transition from $A$ is to $B$, with intensity $0.10$ per year and a transition benefit of $\$500$. In state $B$, the premium rate is zero, the expense rate is $\$250$ per year, and the continuous benefit rate is $\$800$ per year. The transition from $B$ to state $C$ has intensity $0.20$, with $V_C=\$6{,}000$ and transition benefit $\$200$. Find $V_A'$ and $V_B'$.

Method. Apply the Thiele equation separately in each current state. The premium in $A$ is not included in the equation for $B$, but the stated expenses and benefits in $B$ are included.

For state $A$, the interest term is $0.04(12{,}000)=\$480$. The continuous-flow contribution is $1{,}200-180-300=\$720$. The transition contribution is $0.10(12{,}000-9{,}000-500)=0.10(2{,}500)=\$250$. Therefore,

V_A'=480+720+250=\$1{,}450\text{ per year}.

For state $B$, the interest term is $0.04(9{,}000)=\$360$. There is no premium income, so the continuous-flow contribution is $0-250-800=-\$1{,}050$. The transition contribution is $0.20(9{,}000-6{,}000-200)=0.20(2{,}800)=\$560$. Thus,

V_B'=360-1{,}050+560=-\$130\text{ per year}.

Check. State $B$'s zero premium has not removed its expense or benefit outgo. The negative derivative is consistent with its net local outgo after interest and the transition contribution are included. The common mistake avoided is carrying state $A$'s premium into state $B$.

Worked Example: Premiums Stop During Disability While Expenses Continue

A hypothetical three-state contract has active state $A$, disabled state $D$, and recovery state $R$. Use $\delta=0.02$ per year. At the time considered, $V_A=\$11{,}000$, $V_D=\$9{,}000$, and $V_R=\$7{,}000$. In $A$, premiums are $\$900$ per year, expenses are $\$150$ per year, and there is a transition to $D$ at intensity $0.08$ with a $\$300$ transition benefit. In $D$, no premium is payable; expenses are $\$200$ per year and the continuous disability benefit is $\$600$ per year. Recovery from $D$ occurs at intensity $0.10$, with no transition benefit. In $R$, premiums are $\$400$ per year, expenses are $\$100$ per year, and continuous benefits are $\$100$ per year. The transition from $R$ to $A$ has intensity $0.03$ and no transition benefit. Calculate all three reserve derivatives.

Method. Write the continuous-flow terms from the current state first, then add the applicable transition term.

In state $A$, the interest term is $0.02(11{,}000)=\$220$, and the continuous-flow contribution is $900-150=\$750$. The transition contribution is $0.08(11{,}000-9{,}000-300)=0.08(1{,}700)=\$136$. Hence,

V_A'=220+750+136=\$1{,}106\text{ per year}.

In state $D$, the premium is zero. The interest term is $0.02(9{,}000)=\$180$, and continuous outgo is $200+600=\$800$. The recovery contribution is $0.10(9{,}000-7{,}000)=\$200$. Thus,

V_D'=180-200-600+200=-\$420\text{ per year}.

In state $R$, the interest term is $0.02(7{,}000)=\$140$, and the continuous-flow contribution is $400-100-100=\$200$. The transition contribution is $0.03(7{,}000-11{,}000)=-\$120$. Therefore,

V_R'=140+200-120=\$220\text{ per year}.

Check. The disabled-state calculation uses no premium, but it retains both the expense and disability benefit. The common mistake avoided is treating “premiums stop” as if all cash flows stop. The separate rates yield distinct equations for the three states even though the contract and valuation time are shared.

Worked Example: Solving for a State-Specific Premium Rate

A hypothetical contract is in state $A$, with $\delta=0.03$, reserve $V_A=\$15{,}000$, expense rate $\$250$ per year, and continuous benefit rate $\$450$ per year. It can transition only to state $B$, where $V_B=\$12{,}000$. The transition intensity is $0.08$ per year, and the transition benefit is $\$1{,}000$. Find the premium rate in $A$ that makes the instantaneous reserve derivative zero.

Method. Treat the premium rate $p_A$ as the unknown in the Thiele equation. The interest term is $0.03(15{,}000)=\$450$. The transition contribution is $0.08(15{,}000-12{,}000-1{,}000)=0.08(2{,}000)=\$160$. Substitute these values and set the derivative to zero:

$$ 0=450+p_A-250-450+160. $$

The known terms sum to $450-250-450+160=-90$, so $0=p_A-90$, giving $p_A=\$90$ per year. Substitution checks the result: $450+90-250-450+160=0$. The common mistake avoided is omitting the transition benefit from the transition term; doing so would overstate the transition contribution by $0.08(1{,}000)=\$80$ per year and produce the wrong premium.

Common Mistakes and Scope of the Formula

The state-specific equation is a local differential relationship, not a claim that every state has the same premium or expense structure. In particular, a premium can stop on entering a state while contract expenses continue. The rate $p_j$ must reflect the premium rule in state $j$; it should not be inferred from a premium payable in another state.

Using a premium from the wrong state. If premiums are payable only in the active state, set $p_j=0$ in a disabled state.
Removing expenses with premiums. Expenses remain in the equation whenever they are incurred in that state, regardless of whether premium income is payable.
Reversing a transition difference. With a transition benefit paid on $j$ to $k$, the contribution is $\mu_{jk}(V_j-V_k-b_{jk})$, not $\mu_{jk}(V_k-V_j-b_{jk})$.
Mixing units. If intensities are per year, the reserve derivative and continuous rates must also be expressed per year.

The formula applies when the contract has well-defined state-conditional reserves, the transition intensities describe the modeled process, and the specified continuous cash flows and transition payments follow the stated timing convention. If a cash flow occurs only at a transition, it belongs in the corresponding $b_{jk}$, not in the continuous rate $c_j$. Boundary or terminal reserve conditions are supplied separately; this equation describes the reserve evolution between such conditions.

Key takeaway: Construct each row of the multi-state Thiele system from the state currently occupied: use that state's premium, expense, and continuous-benefit rates, then add a transition term for every possible destination. A zero premium rate does not imply zero expense or benefit rates.

Check Your Understanding

For each question, use the state-specific cash flows and the transition convention defined above.

A hypothetical state $B$ has reserve $\$8{,}000$, $\delta=0.025$, no premium, expenses of $\$120$ per year, and continuous benefits of $\$280$ per year. Its only transition is to $C$, with intensity $0.15$, $V_C=\$6{,}500$, and transition benefit $\$300$. Find $V_B'$. (Answer: $-\$20$ per year)
A policy in active state $A$ has reserve $\$10{,}000$, $\delta=0.03$, premium rate $\$700$, and expense rate $\$100$. It has no continuous benefit and transitions to disabled state $D$ at intensity $0.05$, where $V_D=\$9{,}000$; the transition benefit is $\$200$. Find $V_A'$. (Answer: $\$940$ per year)
A hypothetical care state has reserve $\$15{,}000$, $\delta=0.04$, no premium, expenses of $\$350$ per year, and continuous benefits of $\$900$ per year. It transitions to recovery at intensity $0.10$, where the reserve is $\$11{,}000$, and pays a $\$500$ transition benefit. Find the reserve derivative in the care state. (Answer: $-\$300$ per year)
A state $A$ has reserve $\$12{,}000$, $\delta=0.02$, expenses of $\$160$ per year, and continuous benefits of $\$280$ per year. It transitions to $B$ at intensity $0.08$, with $V_B=\$9{,}000$ and transition benefit $\$1{,}000$. What premium rate in $A$ makes $V_A'=0$? (Answer: $\$40$ per year)
A disabled state has reserve $\$10{,}000$, $\delta=0.03$, no premium, expenses of $\$200$ per year, and continuous benefits of $\$500$ per year. It transitions to recovery at intensity $0.10$, where the reserve is $\$8{,}000$, with no transition benefit. A premium of $\$600$ is payable in recovery only. Find the correct disabled-state derivative and identify the error in including the recovery premium. (Answer: $-\$200$ per year; including the recovery premium incorrectly gives $\$400$ per year)

What You'll Learn