Expense Changes and Their Signature Effects

Why Expense Timing Changes the Profit Signature

A change in expenses does not usually affect every duration equally. A one-time increase in a renewal expense changes profit only in the duration when it is incurred. An increase that applies at every renewal changes several entries of the profit signature. In either case, the effect on the expected profit per original policy depends on the chance that the policy is still in force when the expense is paid.

The timing convention matters. This tutorial treats renewal expenses as paid at the end of a policy year, provided the policy is in force at that time. The initial expense is held fixed, so it is not part of the changes considered here. If a product instead pays an expense at the beginning of a duration, the probability of being in force at that payment time must be used; it is not automatically the same as the probability of surviving to the end of that duration.

Definition: Let $Z_t$ be the random profit attributable to duration $t$ for one policy issued at time 0, and let $\pi_t=E[Z_t]$ be its expected duration-$t$ profit, or expected profit signature entry. Suppose a deterministic renewal expense $e_t$ is paid at the end of duration $t$ if the policy is then in force. If that expense changes by $\Delta e_t$, with positive values denoting an increase, its effect on the expected signature is $\Delta\pi_t=-{}_tp\,\Delta e_t$, where ${}_tp$ is the probability the policy is in force at the end of duration $t$. All other cash flows are held fixed.

Under an effective annual interest rate $i$, let $v=1/(1+i)$. With profits arising at duration ends, the change in expected NPV is

\Delta\operatorname{NPV} =\sum_{t=1}^{n}v^t\Delta\pi_t =-\sum_{t=1}^{n}v^t\,{}_tp\,\Delta e_t.

A positive expense increase therefore creates a nonpositive change in profit and NPV. An expense reduction has the opposite sign. These formulas concern the expected profit signature per original policy; they do not treat the expense as though it were incurred by every policy originally issued.

Why Survival and Discounting Both Appear

The survival factor enters because the expense is paid only if the policy is in force at the payment time. The discount factor enters because the resulting profit change occurs at the end of that duration. They answer different questions: survival weights the chance of payment, while discounting converts the amount at that future time to a time-0 value.

Represent the expense change.
At duration $t$, the additional expense is $\Delta e_t$ if the policy is in force at the end of the year, and zero otherwise.

Take its expectation.
The expected additional expense per original policy is ${}_tp\,\Delta e_t$. Because an expense reduces profit, the expected profit change is its negative.

Discount the profit change.
A duration-$t$ amount is multiplied by $v^t$, so its contribution to the NPV change is $-v^t{}_tp\,\Delta e_t$.

For example, suppose each year has the same conditional probability $r$ of remaining in force, with no lapses and annual death probability $1-r$. Then ${}_tp=r^t$. If a constant expense increase $\Delta e$ applies at each year-end from duration $m$ through duration $n$, its NPV effect is

\Delta\operatorname{NPV} =-\Delta e\sum_{t=m}^{n}(rv)^t =-\Delta e\,(rv)^m \frac{1-(rv)^{n-m+1}}{1-rv}, \qquad rv\ne 1.

The first equality adds the discounted expected effects duration by duration. The second is the finite geometric-series formula. It applies only when the same expense change, continuation probability and interest rate apply throughout the stated durations. If any of these vary, calculate and sum the individual terms instead.

Worked Example: A One-Time Renewal-Expense Increase

A hypothetical three-year policy has an annual death probability of 0.10, independently from year to year, and no lapses. Its renewal expense is increased by $12 at the end of duration 2 only, and the expense is paid only if the policy is then in force. Use an effective annual interest rate of 5%. The expense change is deterministic per policy in force; the only random event is death, with annual probability 0.10. Find the signature change and NPV change per policy issued.

Method. Apply the end-of-duration survival probability to the expense change, then discount the expected profit effect to time 0. The continuation probability is $r=1-0.10=0.90$, so the probability of being in force at the end of duration 2 is $0.90^2=0.81$.

At duration 2, the expected additional expense per original policy is $12(0.81)=\$9.72$. The expected profit signature changes by its negative:

\Delta\pi_2=-12(0.90)^2=-\$9.72.

There are no expense changes at the other durations, so $\Delta\pi_1=0$ and $\Delta\pi_3=0$. Here $v=1/1.05$, and the NPV change is

\Delta\operatorname{NPV} =v^2\Delta\pi_2 =-\frac{9.72}{(1.05)^2} =-\$8.816327 \approx-\$8.82.

The result is a reduction of $8.82 per policy issued, rounded to the nearest cent. The signature change is larger in magnitude than the NPV change because it is expressed at duration 2 rather than time 0. The check is that an increased expense must reduce both expected profit at the affected duration and NPV.

Recurring Changes Affect Several Signature Entries

A recurring renewal-expense change produces a sequence of effects, not a single lump at the first affected duration. The conditional expense change per policy in force may be the same each year, but the expected effect per original policy can vary because fewer policies remain in force at later durations. Discounting adds a further duration-dependent weight.

Worked Example: A Recurring Increase Over Three Durations

A hypothetical four-year policy has annual death probability 0.20, independently from year to year, and no lapses. A renewal expense increases by $3 at the end of durations 2, 3 and 4. The increase is deterministic per policy in force, with no expense variability. The effective annual interest rate is 25%. Find each expected signature change and the total NPV change per policy issued.

Method. The annual continuation probability is $r=0.80$. For each affected duration $t$, multiply the $3 increase by $r^t$ to obtain the expected expense per original policy, reverse the sign for profit, and then multiply by $v^t$, where $v=1/1.25=0.80$.

At duration 2, the probability of being in force is $0.80^2=0.64$, so $\Delta\pi_2=-3(0.64)=-\$1.92$. Its discounted contribution is $-1.92(0.80)^2=-\$1.2288$.

At duration 3, the probability of being in force is $0.80^3=0.512$, so $\Delta\pi_3=-3(0.512)=-\$1.536$. Its discounted contribution is $-1.536(0.80)^3=-\$0.786432$.

At duration 4, the probability of being in force is $0.80^4=0.4096$, so $\Delta\pi_4=-3(0.4096)=-\$1.2288$. Its discounted contribution is $-1.2288(0.80)^4=-\$0.50331648$.

Adding the three discounted effects gives

\Delta\operatorname{NPV} =-1.2288-0.786432-0.50331648 =-\$2.51854848 \approx-\$2.52.

Thus the signature changes are $-\$1.92$, $-\$1.536$, and $-\$1.2288$ at durations 2, 3, and 4, respectively, while the NPV decreases by $2.52 per policy issued, rounded to the nearest cent. The decreasing expected signature effects are consistent with the decreasing probabilities of being in force.

Expense Reductions Can Offset Later Increases

The same calculation can compare changes at different durations. An early expense reduction can offset later expense increases in NPV, but the amount of the reduction must account for the probability of being in force when each expense is paid and the relevant discount factors. Equal dollar changes at different durations do not generally have equal NPV effects.

Worked Example: Finding an Early Expense Reduction That Offsets Later Increases

A hypothetical three-year policy has annual death probability 0.10, independently from year to year, and no lapses. There is no interest. Renewal expenses increase by $4 at the ends of durations 2 and 3. What reduction $D$ in the end-of-duration-1 expense makes the total expected NPV change zero? Each expense change is deterministic per policy in force.

Method. With no interest, NPV is the sum of the expected duration effects. The continuation probability is $r=0.90$. The increase at duration 2 reduces expected profit by $4(0.90)^2=\$3.24$; at duration 3 it reduces expected profit by $4(0.90)^3=\$2.916$. Together, the increases reduce NPV by $\$3.24+\$2.916=\$6.156$.

A reduction of $D$ at the end of duration 1 increases profit by $D$ if the policy is in force at that time. Its expected profit increase per original policy is $0.90D$. To make the total change zero, equate this increase to the later expected losses:

0.90D=6.156, \qquad D=\frac{6.156}{0.90}=\$6.84.

The check is $0.90(6.84)=\$6.156$, exactly offsetting $4(0.90)^2+4(0.90)^3=\$6.156$. Therefore the required end-of-duration-1 expense reduction is $6.84 per policy in force.

Common Timing and Weighting Traps

A frequent error is using the probability of being in force at the start of a duration when the problem says the expense is paid at its end. Under the end-of-duration convention used here, the payment requires the policy to remain in force through that year. Another error is to weight for survival but forget to discount, or to discount but omit the survival probability. Keeping the steps separate prevents both mistakes.

For instance, suppose the annual continuation probability is 0.80, a $10 expense increase is paid at the end of duration 3, and $i=25\%$. A calculation using the start-of-duration probability would be

-10(0.80)^2(0.80)^3=-\$3.2768,

which is wrong for an expense paid only if the policy is in force at the end of duration 3. It uses $0.80^2$ instead of the end-of-duration probability $0.80^3$. The correct calculation is

\Delta\pi_3=-10(0.80)^3=-\$5.12, \qquad \Delta\operatorname{NPV} =-10(0.80)^3(0.80)^3 =-\$2.62144.

The first factor $0.80^3$ in the NPV calculation is the probability of being in force at the expense payment time. The second is the discount factor $v^3$. The timing convention should always be read before selecting the probability and discount exponent.

Key takeaway: For an end-of-duration renewal expense, first multiply the per-policy expense change by the probability of being in force at that duration’s end, then reverse the sign to obtain its profit effect. Discount that expected effect by $v^t$ to obtain its NPV contribution. Sum across all durations where the expense changes.

Check Your Understanding

For each problem, treat the stated expense as payable at the end of the specified duration only if the policy is then in force. Assume no lapses and independent annual death outcomes unless stated otherwise.

A hypothetical three-year policy has annual death probability 0.20 and no interest. A $5 renewal expense increase applies only at the end of duration 2. What is the NPV change per policy issued? (Answer: $-\$3.20$)
A hypothetical four-year policy has annual death probability 0.10 and no interest. The end-of-year renewal expense increases by $2 at durations 2, 3 and 4. Find the total NPV change per policy issued. (Answer: $-\$4.3902$)
A hypothetical policy has annual death probability 0.10, no lapses, and $i=10\%$. A $20 expense increase applies only at the end of duration 3. Find its NPV effect per policy issued, to four decimal places. (Answer: approximately $-\$10.9542$)
A hypothetical five-year policy has annual death probability 0.20, no lapses, and $i=25\%$. A $2 renewal expense increase applies at each year-end from duration 2 through duration 5. Find the NPV change per policy issued, to four decimal places. (Answer: approximately $-\$1.8938$)
A hypothetical three-year policy has annual death probability 0.10, no lapses, and no interest. Renewal expenses increase by $6 at the ends of durations 2 and 3. What reduction in the end-of-duration-1 expense exactly offsets those increases in NPV? (Answer: a duration-1 expense reduction of $\$10.26$)

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