Expense-Loaded Single-State Thiele Equations

Why Expense Timing Matters

A reserve must reflect the future cash flows promised or expected under the contract, including expenses. In a single-state life insurance model, a maintenance expense is incurred continuously while the policy remains in force. A death expense, such as a hypothetical claim-settlement cost, is incurred only if death occurs. These expense patterns have different effects on the reserve: the first accumulates over time in force, while the second is weighted by the force of mortality.

This tutorial uses a single insured state, with death as the only exit. Interest and mortality forces are constant over the term considered, and premiums and maintenance expenses are paid continuously while the policy is in force. These assumptions produce a closed-form solution; the formula should not be used unchanged when forces or cash-flow rates vary with time.

Definition: Let $V_t$ be the prospective reserve at time $t$ while the insured is alive. Let $\delta$ be the force of interest, $\mu$ the force of mortality, $p$ the continuous premium rate, $c$ the continuous maintenance expense rate, $B$ the death benefit, and $e$ the expense payable at death. The total payment on death is $B+e$. The symbols $p$ and $c$ are amounts per year while the policy is in force.

The reserve convention is that death at time $t$ releases the reserve $V_t$ and requires payment of $B+e$. Thus the net cash outgo at death, after allowing for the reserve released, is $B+e-V_t$. A continuous expense rate is not a death expense, and an expense payable at death is not paid continuously.

The Expense-Loaded Single-State Equation

Over a short interval of length $dt$, the reserve earns interest at rate $\delta$, receives premium income at rate $p$, and incurs maintenance expenses at rate $c$. Death occurs with probability $\mu\,dt+o(dt)$. Conditional on death, the net outgo is $B+e-V_t$, because the reserve is released. This gives the single-state Thiele equation:

\frac{dV_t}{dt} =\delta V_t+p-c-\mu(B+e-V_t) =(\delta+\mu)V_t+p-c-\mu(B+e).

The death expense appears inside the death-contingent payment and is therefore multiplied by $\mu$. The continuous maintenance expense appears directly as $c$, since it is incurred per unit time while the policy remains in force.

Key result: If $\delta$, $\mu$, $p$, $c$, $B$, and $e$ are constant on $[t,T]$, and $V_T=R$, define $k=\delta+\mu$. Then $$ V_t=R e^{-k(T-t)} +\frac{c+\mu(B+e)-p}{k}\left(1-e^{-k(T-t)}\right), \qquad k\ne 0. $$ Here $R$ is the reserve boundary at $T$, after any terminal cash flow included in the contract has been handled. For a contract ending with no further liability, the usual boundary is $R=0$.

Why the Closed-Form Formula Holds

The equation is a first-order linear differential equation. Write $q=p-c-\mu(B+e)$, so $V'_t=kV_t+q$. Multiply both sides by the integrating factor $e^{-kt}$. The product rule gives the derivative on the left:

\frac{d}{dt}\left(e^{-kt}V_t\right)=q e^{-kt}.

Integrate from $t$ to $T$, using $V_T=R$. Since $k$ is constant, the integral of $e^{-ks}$ from $t$ to $T$ is $(e^{-kt}-e^{-kT})/k$. Rearranging yields:

e^{-kT}R-e^{-kt}V_t =\frac{q}{k}\left(e^{-kt}-e^{-kT}\right), $$ $$ V_t=R e^{-k(T-t)} -\frac{q}{k}\left(1-e^{-k(T-t)}\right).

Substituting $q=p-c-\mu(B+e)$ gives the stated formula. This form also makes the expense effects visible: all else equal, increasing $c$ by one dollar per year increases $V_t$ by $(1-e^{-k(T-t)})/k$, while increasing $e$ by one dollar increases it by $\mu(1-e^{-k(T-t)})/k$. The second effect includes the chance of death; it is not the same as adding one dollar of continuous expense.

Worked Calculations

Worked Example: Separating Maintenance and Death Expenses

A hypothetical 8-year term policy has constant force of interest $\delta=0.03$ and constant force of mortality $\mu=0.02$. It pays a death benefit of $B=\$8{,}000$ and incurs an additional $\$200$ expense at death. The continuous premium rate is $p=\$100$ per year and the continuous maintenance expense rate is $c=\$40$ per year. There is no liability at the end of year 8, so $V_8=0$. Find $V_0$ and the increase caused by the death expense.

Method. Apply the constant-force formula with $T=8$, $R=0$, and $k=\delta+\mu$. This keeps the death expense conditional on death rather than treating it as a yearly expense.

First, $k=0.03+0.02=0.05$, and the total payment on death is $B+e=\$8{,}200$. Its expected outgo rate while in force is $\mu(B+e)=0.02(8{,}200)=\$164$ per year. Therefore the net amount $c+\mu(B+e)-p$ is $40+164-100=\$104$ per year.

The time factor is $1-e^{-kT}=1-e^{-0.05(8)}=1-e^{-0.4}=1-0.670320046=0.329679954$. Thus $(1-e^{-kT})/k=0.329679954/0.05=6.59359908$.

The reserve is $V_0=104(6.59359908)=\$685.7343$, which rounds to \$685.73 to the nearest cent.

To isolate the death-expense effect, its contribution is $\mu e(1-e^{-kT})/k=0.02(200)(6.59359908)=\$26.3744$, or \$26.37 to the nearest cent. Without the death expense, the reserve would be $685.7343-26.3744=\$659.3599$, which rounds to $\$659.36$. Using unrounded reserves, the difference is $\$26.3744$, which rounds to $\$26.37$.

Check and common mistake avoided. The death expense increases the reserve, as an additional future outgo should. Treating $\$200$ as a continuous annual expense would incorrectly add $200(6.59359908)=\$1{,}318.72$; the correct rate contribution is $\mu e=0.02(200)=\$4$ per year in the Thiele equation.

Worked Example: Reserve With a Nonzero Boundary

A hypothetical contract is valued at time 0 for a 10-year period. Assume constant forces $\delta=0.02$ and $\mu=0.03$, continuous premium rate $p=\$250$ per year, maintenance expense rate $c=\$60$ per year, death benefit $B=\$12{,}000$, and death expense $e=\$1{,}000$. The reserve boundary is $V_{10}=\$500$. Calculate $V_0$, including the separate contribution of the death expense.

Method. Use the terminal-boundary form of the solution. The boundary value must be discounted back as well as the net future outgo accumulated over the term.

Here $k=0.02+0.03=0.05$, $B+e=\$13{,}000$, and $\mu(B+e)=0.03(13{,}000)=\$390$ per year. Hence $c+\mu(B+e)-p=60+390-250=\$200$ per year. As above, $e^{-0.5}=0.606530660$ and $(1-e^{-0.5})/0.05=7.86938680$.

The discounted boundary contribution is $500(0.606530660)=\$303.2653$. The net-outgo contribution is $200(7.86938680)=\$1{,}573.8774$. Therefore $V_0=303.2653+1{,}573.8774=\$1{,}877.1427$, which rounds to \$1,877.14.

The death expense alone contributes $\mu e(1-e^{-kT})/k=0.03(1{,}000)(7.86938680)=\$236.0816$, or \$236.08 to the nearest cent. If the death expense were omitted, the reserve would be $\$1{,}877.1427-\$236.0816=\$1{,}641.0611$, or $\$1{,}641.06$ to the nearest cent.

Check and common mistake avoided. The terminal reserve contributes $\$303.27$, not $\$500$, at time 0 because it is discounted over 10 years at force $k=\delta+\mu$ in this backward reserve equation. Omitting the boundary term would understate the result.

Worked Example: Valuing Expenses From an Intermediate Duration

For a hypothetical 12-year policy, assume constant forces $\delta=0.03$ and $\mu=0.02$. The continuous premium rate is $\$180$ per year, the maintenance expense rate is $\$55$ per year, and the death benefit is $\$15{,}000$. An additional $\$750$ expense is payable at death. The reserve boundary at year 12 is $V_{12}=\$1{,}000$. Find the reserve at year 4 and quantify how much of it is attributable to the death expense.

Method. Apply the formula over the remaining duration, $12-4=8$ years, using the boundary at year 12. Do not use the full 12-year term when valuing from year 4.

The combined force is $k=0.03+0.02=0.05$. The total death payment is $15{,}000+750=\$15{,}750$, so its expected outgo rate is $0.02(15{,}750)=\$315$ per year. The net outgo rate after premium income is $c+\mu(B+e)-p=55+315-180=\$190$ per year.

For the remaining 8 years, $e^{-0.05(8)}=e^{-0.4}=0.670320046$, and $(1-e^{-0.4})/0.05=6.59359908$. The discounted boundary contribution is $1{,}000(0.670320046)=\$670.3200$. The net-outgo contribution is $190(6.59359908)=\$1{,}252.7838$. Thus $V_4=\$1{,}923.1038$, or \$1,923.10 to the nearest cent.

The death expense contributes $0.02(750)(6.59359908)=\$98.9040$, or \$98.90. Removing that expense would give a reserve of $\$1{,}923.1038-\$98.9040=\$1{,}824.1998$, or $\$1{,}824.20$ to the nearest cent.

Check and common mistake avoided. The death-expense contribution is positive and less than the expense amount because it is both contingent on death and discounted over the remaining term. The eight-year remaining duration, not the original 12-year duration, governs both factors.

Interpreting Expense Changes and Avoiding Traps

Under the constant-force assumptions, the effect of an expense change can be read directly from the solution. For a remaining term $n=T-t$, an increase $\Delta c$ in the annual maintenance expense rate changes the reserve by $\Delta c(1-e^{-kn})/k$. An increase $\Delta e$ in the expense payable at death changes it by $\mu\Delta e(1-e^{-kn})/k$. Both are positive for positive expense increases, but their multipliers differ.

Expense change	Change in the reserve at time $t$	Reason
Increase $c$ by $\Delta c$	$\Delta c(1-e^{-kn})/k$	Incurred continuously while the policy is in force
Increase $e$ by $\Delta e$	$\mu\Delta e(1-e^{-kn})/k$	Incurred only when death occurs

A frequent exam error is to put the death expense in the continuous-expense term, writing $p-(c+e)$ instead of $p-c-\mu(B+e)$. That treats the full death expense as payable every year. A second error is to omit the reserve release at death and use $-\mu(B+e)$ without the $+\mu V_t$ term. That changes the differential equation and therefore the discounting factor from $\delta+\mu$ to $\delta$, which is inconsistent with the stated reserve convention.

Key takeaway: A continuous maintenance expense enters the Thiele equation directly as an annual rate. An expense payable on death belongs with the death benefit and is multiplied by the force of mortality. Under constant forces, apply the resulting closed-form formula only over the period for which those forces and cash-flow rates remain constant.

Check Your Understanding

For each problem, use the stated single-state assumptions and round money to the nearest cent. The answer is given so you can check your calculation.

A hypothetical 5-year policy has $\delta=0.04$, $\mu=0.01$, terminal reserve zero, and all benefits and premiums unchanged. By how much does the reserve at time 0 increase if the continuous maintenance expense rate rises by $\$30$ per year? (Answer: $\$132.72$)
For a 6-year term, $\delta=0.04$, $\mu=0.015$, and $V_6=0$. Find the increase in $V_0$ when a death expense of $\$400$ is added, with other cash flows unchanged. (Answer: $\$30.66$)
A 5-year policy has $\delta=0.02$, $\mu=0.03$, continuous premium rate $\$120$, maintenance expense rate $\$25$, death benefit $\$3{,}000$, death expense $\$500$, and terminal reserve zero. Find $V_0$. (Answer: $\$44.24$)
A 10-year policy has $\delta=0.03$, $\mu=0.02$, continuous premium rate $\$250$, maintenance expense rate $\$40$, death benefit $\$10{,}000$, death expense $\$1{,}000$, and terminal reserve zero. Find $V_0$. (Answer: $\$78.69$)
For an 8-year remaining term, $\delta=0.025$ and $\mu=0.015$. Other assumptions, including the terminal reserve, are unchanged. By how much does the reserve increase if maintenance expenses rise from $\$20$ to $\$35$ per year? (Answer: $\$102.69$)

Expense change	Change in the reserve at time \(t\)	Reason
Increase \(c\) by \(\Delta c\)	\(\Delta c(1-e^{-kn})/k\)	Incurred continuously while the policy is in force
Increase \(e\) by \(\Delta e\)	\(\mu\Delta e(1-e^{-kn})/k\)	Incurred only when death occurs

What You'll Learn