Expense Assumption Changes and Reserve Effects

Why Expense Assumptions Affect Both Reserves and Profit

An expense assumption change can affect several parts of a long-term product projection at once. Higher maintenance expenses increase expected future outgo, so prospective reserves generally rise. If the expenses actually incurred differ from the basis used to calculate reserves, the difference also changes annual profit. The timing of the expenses and the policyholder’s state at each payment date determine the size and pattern of those effects.

The calculations below use a hypothetical contract with active and disabled states and death as a terminating state. The discrete examples use annual expenses paid at the beginning of each year, transition benefits paid at year-end, and an effective annual interest rate. The continuous-time section states its assumptions separately. This distinction matters: changing the payment timing changes the reserve and profit formulas.

Definition: A prospective reserve is the expected present value of future benefits and expenses less the expected present value of future premiums, conditional on the policy being in a specified state at the valuation time. An expense assumption change is the difference between the revised and original assumptions for expenses by amount, time, or state.

Discrete Reserves with Start-of-Year Expenses

Suppose that at time $k$, immediately before the year’s premium and expense payments, the policy is in state $s$. Let $V_k^s$ be the reserve, $P_k^s$ the premium received at that time, and $e_k^s$ the expense paid then. Let $p_{sr}$ be the probability of being in state $r$ at year-end, conditional on state $s$ at the start of the year. Let $b_{sr}$ be the benefit paid at year-end on that transition. A terminating state has reserve zero.

Under a prospective valuation basis with effective annual interest rate $i$, the recursion is

V_k^s=e_k^s-P_k^s+\frac{\sum_r p_{sr}(b_{sr}+V_{k+1}^r)}{1+i}.

The current expense and premium are immediate, so they are not discounted. The benefit and next reserve are paid or held at year-end, so they are discounted one year. This timing convention is used throughout the discrete examples.

If premiums, benefits, transitions, and interest assumptions stay fixed, a change in the expense assumptions changes the reserve by the present value of the added future expenses. In the recursion, the change in current expense is added immediately; changes in next-year reserves are probability-weighted and discounted.

Worked Example: Revaluing State-Specific Expenses

A hypothetical two-year contract has active state $A$, disabled state $D$, and death as a terminating state. The effective annual interest rate is $i=0.05$. At the start of each year, an active policy pays a premium of $30; a disabled policy pays no premium. Start-of-year expenses are $10 in $A$ and $15 in $D$. At year-end, a $40 benefit is paid if the policy is alive and disabled, and a $200 benefit is paid on death. The one-year transition probabilities are $A\to A=0.8$, $A\to D=0.1$, $A\to\text{death}=0.1$, $D\to A=0.2$, $D\to D=0.7$, and $D\to\text{death}=0.1$. There are no benefits on recovery. The reserve at the end of year 2 is zero. Find the reserves under the original expenses, then find the reserve increases if annual expenses rise by $4 in $A$ and $3 in $D$.

Method. Work backward from the terminal reserves. The expected year-end benefit in $A$ is $0.1(40)+0.1(200)=24$. In $D$, it is $0.7(40)+0.1(200)=48$.

At time 1, the next reserves are zero. Therefore,

V_1^A=10-30+\frac{24}{1.05}=2.8571, \qquad V_1^D=15+\frac{48}{1.05}=60.7143.

At time 0, substitute these time-1 reserves into the recursion:

\begin{aligned} V_0^A &=10-30+\frac{24+0.8(2.8571)+0.1(60.7143)}{1.05}\\ &= -20+\frac{32.3571}{1.05}=10.8163,\\ V_0^D &=15+\frac{48+0.2(2.8571)+0.7(60.7143)}{1.05}\\ &=15+\frac{91.0714}{1.05}=101.7347. \end{aligned}

Now calculate the reserve increases. At time 1, no future years remain, so the added expense increases the reserve by exactly the added current expense: $\Delta V_1^A=4$ and $\Delta V_1^D=3$. At time 0,

\begin{aligned} \Delta V_0^A &=4+\frac{0.8(4)+0.1(3)}{1.05} =4+\frac{3.5}{1.05}=7.3333,\\ \Delta V_0^D &=3+\frac{0.2(4)+0.7(3)}{1.05} =3+\frac{2.9}{1.05}=5.7619. \end{aligned}

The revised time-0 reserves are therefore $18.15 in $A$ and $107.50 in $D$, rounded to cents. The added reserve in $A$ also checks against the value of expected added expenses: $4 now, plus an expected $3.50 one year later, discounted at 5%, gives $4+3.50/1.05=7.3333$.

Common mistake avoided: Discounting the current expense would incorrectly give it a factor of $1/1.05$. It is paid at the valuation time, so it enters the reserve recursion without discounting.

Why Expense Changes Enter Thiele Equations

For a continuous-time model, let $V_j(t)$ be the reserve at time $t$ in state $j$. Assume a constant force of interest $\delta$, continuous premium income rate $p_j(t)$, and continuous expense rate $e_j(t)$. Transitions from state $j$ to state $k$ occur with intensity $\mu_{jk}(t)$, and a benefit $b_{jk}(t)$ is paid on that transition. A terminating state has reserve zero. The model assumes these cash-flow rates and transition intensities are well-defined over the period considered.

Over a short interval of length $h$, the probability of one transition $j\to k$ 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 reserve earns interest over the interval, premium income adds value, and expenses subtract value. On a transition, the insurer pays the transition benefit and moves to the reserve for the new state. Equating the reserve at the start to the expected discounted value at the end, subtracting $V_j(t)$, dividing by $h$, and taking the limit as $h\to0$ gives Thiele’s differential equation:

\frac{dV_j}{dt} =\delta V_j+p_j-e_j +\sum_{k\ne j}\mu_{jk}\bigl(b_{jk}+V_k-V_j\bigr).

This form uses premium income and expenses as continuous rates and transition benefits as payments at the transition. It does not directly apply to expenses paid only at discrete policy anniversaries; those cash flows must instead be included at their actual payment times.

If only the expense rates change, while interest, premiums, transitions, and benefits remain the same, subtracting the original Thiele equation from the revised one gives

\frac{d\,\Delta V_j}{dt} =\delta\Delta V_j-\Delta e_j+ \sum_{k\ne j}\mu_{jk}(\Delta V_k-\Delta V_j),

where $\Delta V_j$ is the revised reserve less the original reserve and $\Delta e_j$ is the revised expense rate less the original expense rate. The transition-benefit terms cancel because the benefits have not changed. At a terminal time $T$ with no future obligations, the boundary condition is $\Delta V_j(T)=0$ in every non-terminating state.

Worked Example: Interpreting a Thiele Expense Effect

In a hypothetical continuous-time contract, the force of interest is $\delta=0.04$. There are two non-terminating states, active $A$ and disabled $D$, and a terminating death state with $\Delta V_{\text{death}}=0$. At a particular time, the reserve increases from the revised expense basis are $\Delta V_A=6$ and $\Delta V_D=4$. The expense-rate increases are $\Delta e_A=1.5$ and $\Delta e_D=0.5$ per year. Transition intensities are $\mu_{AD}=0.2$, $\mu_{A,\text{death}}=0.1$, $\mu_{DA}=0.1$, and $\mu_{D,\text{death}}=0.15$, all per year. Find the instantaneous rates of change of the reserve increases.

Method. Apply the expense-difference form of Thiele’s equation. In state $A$, the interest contribution is $0.04(6)=0.24$. The transition contribution to $D$ is $0.2(4-6)=-0.40$, and the transition contribution to death is $0.1(0-6)=-0.60$. Thus

\frac{d\,\Delta V_A}{dt} =0.24-1.5-0.40-0.60=-2.26.

In state $D$, the interest contribution is $0.04(4)=0.16$. The transition contribution to $A$ is $0.1(6-4)=0.20$, and the transition contribution to death is $0.15(0-4)=-0.60$. Therefore,

\frac{d\,\Delta V_D}{dt} =0.16-0.5+0.20-0.60=-0.74.

At this instant, the reserve increases are falling at 2.26 per year in $A$ and 0.74 per year in $D$. This is consistent with the zero reserve-increase boundary condition at the end of the contract, although the local derivatives alone do not determine the full reserve paths.

Common mistake avoided: Omitting the $-\mu_{jk}\Delta V_j$ part of each transition term overstates the derivative. A transition removes the policy from its current state as well as placing it in the destination state.

Annual Profit and NPV Under a Changed Expense Basis

Reserve changes do not by themselves specify annual profit: the answer depends on which reserve basis is held and which expenses are actually incurred. For the discrete timing used earlier, an end-of-year profit conditional on starting the year in state $s$ can be written

\Pi_k^s=(1+i)(V_k^s+P_k^s-e_k^s) -\sum_r p_{sr}(b_{sr}+V_{k+1}^r).

If the reserve recursion and the profit calculation use the same assumptions, this expression is zero in expectation, state by state. If actual expenses exceed those used in the held reserve basis by $\Delta e_k^s$, with the other items unchanged, the year-end profit changes by $-(1+i)\Delta e_k^s$. The factor $1+i$ reflects that the start-of-year cash-flow difference is carried to year-end.

Worked Example: Expense-Driven Profit Signature and NPV

Use the two-year contract and transition probabilities from the first worked example. Reserves are held on the original expense basis, while actual start-of-year expenses are higher by $4 in $A$ and $3 in $D$ in each year. Starting in $A$ at time 0, calculate the expected annual profit changes and their NPV at 5%. Assume no expense change after the end of year 2.

Method. The first-year expense increase is $4 with certainty. The end-of-year-1 profit change is therefore $-1.05(4)=-$4.20$. At time 1, the probabilities of being in $A$, $D$, and death are 0.8, 0.1, and 0.1. The expected second-year expense increase is $0.8(4)+0.1(3)=\$3.50$; death has no second-year expense. The end-of-year-2 profit change is $-1.05(3.50)=-\$3.675$.

Discount the end-of-year profits to time 0:

\operatorname{NPV} =\frac{-4.20}{1.05}+\frac{-3.675}{1.05^2} =-4.00-3.3333=-\$7.33.

The same result follows directly from the added expense cash flows: $4+3.50/1.05=\$7.3333$, so their NPV is a $7.33 loss. The calculation checks that the profit signature and the expense cash flows use consistent dates and discounting.

Common mistake avoided: Discounting the $3.50 expected expense increase as though it occurred at time 0 places it one year too early. It is paid at the start of year 2, time 1, and is discounted once to time 0.

Practical Checks for Integrated Expense Changes

For a multi-state projection, an expense assumption should be attached to the state and time at which it applies. A change in active-state expenses does not apply during disabled years unless the contract’s expense rule says it does. Likewise, a per-policy expense incurred only while the contract is in force must not be projected after death or another terminating event.

A reliable calculation separates three questions:

When is the expense paid?
Identify whether it is immediate, paid at a policy anniversary, or continuous. This determines whether it is discounted and whether it earns interest before profit recognition.

In which state is it incurred?
Apply the state-specific expense only to policies occupying that state at the payment time, using the corresponding state probabilities or transition intensities.

Which basis determines the reserve?
Keep the actual expense assumption distinct from the expense assumption underlying held reserves. A mismatch produces profit emergence; a reserve revaluation changes the reserve amounts.

The NPV of the resulting profit changes can be checked against the present value of the underlying expense differences when all other cash flows and assumptions are unchanged and terminal liabilities are treated consistently. If the two do not reconcile, inspect expense timing, state occupancy, reserve dates, and whether profits are measured at the start or end of each projection period.

Key takeaway: Higher future expenses generally increase prospective reserves, but the annual profit effect depends on when expenses are paid and whether reserves use the revised basis. Track each expense by state and date, then reconcile the discounted profit signature to the added expense value.

The methods in this tutorial are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated timing and reserve basis carefully.

A single-state contract has three start-of-year expense increases: $2 at time 0, $3 at time 1, and $4 at time 2. The effective annual interest rate is 4%, and no obligations remain after the time-2 payment. Find the reserve increase at time 0. (Answer: $8.58)
At a valuation date, an expense increase of $5 applies immediately. One year later, the reserve increases are $7 in state $A$ and $2 in state $D$; from the current state, the probabilities of those states are 0.6 and 0.2, respectively, with the remaining probability terminating the contract. At $i=0.05$, find the current reserve increase. (Answer: $9.38)
Expenses paid at the start of a year are $6 higher than the expense assumption underlying the held reserve. At an effective annual interest rate of 4%, find the end-of-year profit change, assuming all other cash flows and the reserve are unchanged. (Answer: $-\$6.24$)
A hypothetical projection has expense increases of $8 at time 0, an expected $5 at time 1, and an expected $3 at time 2. All are paid at the start of their respective years. Find their NPV at 5%, rounded to cents. (Answer: $15.48)
In a continuous-time model, $\delta=0.03$, $\Delta V_A=5$, $\Delta V_D=2$, and $\Delta V_{\text{death}}=0$. In state $A$, $\Delta e_A=0.8$, $\mu_{AD}=0.2$, and $\mu_{A,\text{death}}=0.1$. Calculate $d\Delta V_A/dt$. (Answer: $-1.75$ per year)

What You'll Learn