State-Dependent Expenses

Why Expense Timing and State Matter

Expenses are not all incurred for the same reason or at the same time. An insurer might pay an acquisition expense when a policy is issued, incur recurring administration costs while a policy remains in force, and pay an additional claim-handling expense when an insured moves into a claim state. The amount, timing, and state condition attached to each expense determine how it should be valued.

The central task is to describe each expense as a cash flow: specify its amount, when it is paid, and whether payment depends on being in a state or making a transition. Once those details are explicit, its expected present value can be calculated using the relevant state probabilities or transition rates.

Definition: A fixed expense has a specified amount and payment time, such as an acquisition cost paid at issue. A recurring state expense is incurred at specified times, or continuously, while the policy is in a designated state. A transition expense is incurred when a specified transition between states occurs. A fixed amount can still be conditional: for example, a fixed claim-handling amount payable only upon entry to disability is a transition expense.

Representing Expenses in a Multi-State Model

Let the policyholder occupy one of a set of states, such as active, disabled, or dead. In a discrete-time model, write $p_{jk}$ for the probability of being in state $k$ at the next payment time, given state $j$ now. A row of these probabilities sums to 1. If $r_k$ is an expense paid at the end of a period when the policy is in state $k$, then the expected expense for that period is calculated from the probabilities of being in each payment state.

E[\text{expense at time }t] =\sum_k P(\text{state at }t=k)\,r_k

This expression applies when the amount is determined by the state at the payment time. If an expense instead arises on a transition, use the probability of that particular transition during the period, not just the probability of occupying its destination state. The distinction is important: a person can already be disabled without making a new transition into disability in the period being valued.

In continuous time, let $p_k(t)$ be the probability of occupying state $k$ at time $t$, and let $q_{jk}(t)$ be the transition intensity from state $j$ to a different state $k$. For a constant continuous payment rate $c_k$ while in state $k$, the expected expense rate at time $t$ is $\sum_k p_k(t)c_k$. For an expense $e_{jk}$ paid at each $j$-to-$k$ transition, the expected expense rate is $p_j(t)q_{jk}(t)e_{jk}$.

With force of interest $\delta$, a payment at time $t$ is discounted by $e^{-\delta t}$. For an effective annual rate $i$, a payment at integer time $t$ is discounted by $v^t=(1+i)^{-t}$. Use the discounting convention that matches the timing of the cash flow; do not treat a continuous payment as though it were paid only at the year-end.

Valuing Recurring Expenses by State

A state expense is conditional on where the policy is, not on how it arrived there. For discrete annual payments, first calculate the state probabilities at each payment time. Multiply each state probability by the expense assigned to that state, add across states, and discount the resulting expected payment. This method handles different administration costs for active and disabled policies without assigning either cost to a transition.

Worked Example: Annual Maintenance Expenses in Three States

A hypothetical policy begins in the active state $A$. At each policy anniversary, its state follows the same transition matrix, with rows and columns ordered as active $A$, disabled $D$, and dead $M$:

P= \begin{pmatrix} 0.80&0.15&0.05\\ 0.20&0.70&0.10\\ 0&0&1 \end{pmatrix}.

Assume the dead state is absorbing, and the stated transition probabilities include all changes of state over each year. Expenses are paid at the end of each year according to the state then occupied: $20 in $A$, $80 in $D$, and $0 in $M$. There are no acquisition or transition expenses. The effective annual interest rate is $i=5\%$. Find the expected present value of expenses over two years.

Method. Propagate the state probabilities one year at a time, calculate the expected expense at each anniversary, and discount each expected payment to issue.

At time 0 the state probability row vector is $(1,0,0)$. After one year it is:

$$ (1,0,0)P=(0.80,0.15,0.05). $$

The expected expense at the end of year 1 is $0.80(20)+0.15(80)+0.05(0)=16+12=\$28.00$.

After two years, multiply the year-1 probabilities by $P$:

(0.80,0.15,0.05)P =(0.80(0.80)+0.15(0.20),\;0.80(0.15)+0.15(0.70),\;0.80(0.05)+0.15(0.10)+0.05) =(0.67,0.225,0.105).

These probabilities sum to $0.67+0.225+0.105=1$. The expected expense at the end of year 2 is $0.67(20)+0.225(80)+0.105(0)=13.40+18.00=\$31.40$.

The present value is therefore:

\frac{28.00}{1.05}+\frac{31.40}{1.05^2} =26.6667+28.4807 =55.1474.

The expected present value is $55.15, rounded to the nearest cent. The method avoids treating the disabled-state expense as payable to every policyholder who was ever disabled; it is payable only when the policy is disabled at the specified anniversary.

Continuous Expenses and Transition Expenses

The continuous-time formulas follow from considering a short interval of length $h$. Given occupancy of state $j$ at time $t$, the probability of a $j$-to-$k$ transition during that interval is $q_{jk}(t)h+o(h)$, where $o(h)/h$ tends to zero as $h$ tends to zero. Thus the unconditional probability of that transition in the interval is approximately $p_j(t)q_{jk}(t)h$. Multiplying by the expense per transition gives its expected expense in the interval. Discounting to time 0 and summing over intervals gives the integral below.

\text{EPV of }j\text{-to-}k\text{ transition expenses} =\int_0^T e^{-\delta t}p_j(t)q_{jk}(t)e_{jk}\,dt.

The same interval argument gives the expected present value of a continuous expense rate $c_k$ while in state $k$:

\text{EPV of continuous state expenses} =\int_0^T e^{-\delta t}\sum_k p_k(t)c_k\,dt.

These expressions require the payment amounts and rates to be specified as stated, and the state probabilities and intensities to describe the same model. They value expected expenses over the stated horizon; they are not a guarantee of the actual expense on a particular policy.

Worked Example: Continuous Maintenance and Claim-Handling Expenses

Consider a hypothetical policy initially in state $A$ (active). The only possible changes during the first year are $A\to D$, with constant intensity $0.20$ per year, and $A\to M$, with constant intensity $0.10$ per year, where $D$ is disabled and $M$ is dead. Both $D$ and $M$ are absorbing in this model. The holding time in $A$ is therefore exponential with total exit intensity $0.30$. While active, the policy incurs a continuous maintenance expense at rate $40 per year. A $300 expense is paid at an $A\to D$ transition, and a $50 expense is paid at an $A\to M$ transition. There are no expenses after leaving $A$. The force of interest is $\delta=0.20$. In this hypothetical model the effective annual rate is $e^{0.20}-1=0.2214$. Find the expected present value over one year, including a $75 acquisition expense paid at time 0.

Method. The probability of remaining active at time $t$ is the survival probability under the exponential holding time:

p_A(t)=e^{-(0.20+0.10)t}=e^{-0.30t}.

The discounted expected present value of active-state maintenance is:

40\int_0^1 e^{-0.20t}e^{-0.30t}\,dt =40\int_0^1 e^{-0.50t}\,dt =40\frac{1-e^{-0.50}}{0.50} =40(0.7869387) =\$31.4775.

At time $t$, the expected transition expense rate, before discounting, is $p_A(t)[0.20(300)+0.10(50)]$. The bracketed amount is $60+5=65$ dollars per year of active-state exposure. Thus the discounted transition expenses have present value:

65\int_0^1 e^{-0.20t}e^{-0.30t}\,dt =65(0.7869387) =\$51.1510.

The acquisition expense is paid immediately, so its present value is $75.00. Adding the three components gives $75+31.4775+51.1510=\$157.6285$, or $157.63 to the nearest cent. The calculation uses the competing transition intensities separately: it does not assign the disability-transition expense to deaths, or the death-transition expense to disabilities.

Combining Expense Types Without Double Counting

A complete expense valuation is easiest to audit when it is separated into components by both cash-flow type and payment time. An acquisition expense paid at issue is discounted by 1. An annual expense paid at the end of year $t$ is discounted by $v^t$. A transition expense paid at the transition time is discounted at that time, so in continuous time the discount factor remains inside the integral.

The following sequence is useful when several expense provisions apply:

State the trigger.
Identify whether the expense is unconditional at a specified time, conditional on occupying a state, or conditional on a specified transition.

State the payment timing.
Distinguish an expense paid at issue, at an anniversary, continuously during occupancy, or at the instant of transition.

Use the matching probability quantity.
Use occupancy probabilities for state expenses and transition probabilities or intensities for transition expenses.

Discount each component.
Apply the discount factor corresponding to its payment time, then add the component present values.

Worked Example: Acquisition, State, and Transition Expenses

A hypothetical two-state policy starts active, $A$, and can move between active $A$ and disabled $D$. There is no death or other exit during the two-year term. At each year-end, the transition probabilities are $P(A\to A)=0.75$, $P(A\to D)=0.25$, $P(D\to A)=0.20$, and $P(D\to D)=0.80$. The effective annual interest rate is $i=4\%$. A $90 acquisition expense is paid at time 0. At each year-end, a maintenance expense of $12 is paid if active and $30 if disabled after that year's transition. At the same year-end, a transition expense of $200 is paid for $A\to D$, and $60 for $D\to A$. Find the total expected present value.

Method. Value the immediate expense separately. For each year-end, calculate the occupancy-based maintenance expense and the transition-based expense using the probabilities of the relevant events in that year.

At the end of year 1, the state probabilities are $(0.75,0.25)$. Maintenance expense has expectation $0.75(12)+0.25(30)=9+7.50=\$16.50$. The probability of $A\to D$ is $0.25$, and $D\to A$ is impossible because the policy begins in $A$. The expected transition expense is $0.25(200)=\$50.00$. Total expected year-1 expense is $\$66.50$.

At the end of year 2, the state probabilities are:

P(A\text{ at year 2})=0.75(0.75)+0.25(0.20)=0.6125, $$ $$ P(D\text{ at year 2})=0.75(0.25)+0.25(0.80)=0.3875.

They sum to 1. The expected year-2 maintenance expense is $0.6125(12)+0.3875(30)=7.35+11.625=\$18.975$.

For year-2 transitions, the probability of $A\to D$ is $0.75(0.25)=0.1875$, since the policy must be active at the end of year 1 and then move to disabled. The probability of $D\to A$ is $0.25(0.20)=0.05$. The expected transition expense is $0.1875(200)+0.05(60)=37.50+3.00=\$40.50$. Thus the total expected year-2 expense is $18.975+40.50=\$59.475$.

The total present value is:

90+\frac{66.50}{1.04}+\frac{59.475}{1.04^2} =90+63.9423+54.9880 =208.9303.

The answer is $208.93 to the nearest cent. The check on the year-2 probabilities confirms that occupancy costs and transition costs have both been accounted for, but not confused with one another.

Common Mistakes and Why They Matter

A frequent error is to use the probability of being in a destination state as the probability of a transition into that state. In the preceding example, the year-2 probability of being disabled is $0.3875$, but the probability of an $A\to D$ transition during year 2 is only $0.1875$. The remaining disabled policies include those already disabled at the end of year 1 who stay disabled.

For example, incorrectly multiplying $0.3875$ by the $200 $A\to D$ expense would produce a year-2 expense of $77.50 for that transition instead of the correct $37.50. It would overstate the expected transition expense by $40.00 before discounting. The appropriate probability is the joint probability of starting the period in $A$ and moving to $D$, namely $0.75(0.25)$.

Do not omit the state condition. A recurring disabled-state expense is not payable merely because disability can occur; it is payable according to the specified occupancy and payment timing.
Do not charge a transition expense repeatedly for occupancy. A one-time expense on entry to a state is not automatically incurred in every period spent in that state.
Do not discount a time-0 expense. An acquisition expense paid immediately has present value equal to its amount.
Do not mix timing conventions. An end-of-year expense uses the year-end discount factor; a continuous expense uses discounting throughout the period.
Check probability accounting. State probabilities at a payment time should sum to 1, and probabilities of mutually exclusive transitions should be calculated from the states occupied at the start of the interval.

Key takeaway: Value an expense according to its trigger. Occupancy expenses use state probabilities; transition expenses use transition probabilities or intensities; fixed expenses use their specified payment time. Add the discounted components only after each has been calculated on the correct basis.

Check Your Understanding

For each problem, identify the relevant expense timing and use the stated interest and state assumptions.

A hypothetical policy has a $140 expense at issue and pays $40 at each of the next two year-ends, regardless of state. The effective annual interest rate is 5%. Find the expected present value. (Answer: $214.38)
A policy starts active and, over one year, has probability 0.90 of remaining active and 0.10 of entering disability. There is no death or recovery during the year. A year-end expense is $18 if active and $70 if disabled. The effective annual interest rate is 4%. Find the present value of the expected expense. (Answer: $22.31)
A policy starts active. Its only transition is from active to disabled, with constant intensity $0.20$ per year; disability is absorbing. A $250 expense is paid at the transition during the first year. The force of interest is $0.10$. Find the expected present value. (Answer: $43.20)
A two-state policy starts active. At each year-end, $P(A\to D)=0.20$ and $P(D\to A)=0.50$. There is no death. At the end of year 2, a $100 expense is paid for $A\to D$, and a $40 expense for $D\to A$. With zero interest, calculate the expected year-2 transition expense. (Answer: $20.00)
A policy starts active and can move only from active to disabled, at constant intensity $0.20$ per year; disabled is absorbing. There is no death. A continuous maintenance expense of $30 per year is incurred while active over two years. The force of interest is $0.10$. Find its expected present value. (Answer: $45.12)

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