Why Sign Conventions Matter
In a multi-state profit test, a premium received during the year, a claim paid on a transition, and a reserve held for a destination state can all affect profit in the same calculation. A sign error can turn a reserve release into a loss or make an increase in future obligations look like a gain. The reliable approach is to define every amount from the insurer’s perspective and apply one accounting identity consistently.
This tutorial uses annual profit amounts conditional on a specified state or transition. It does not combine those conditional amounts using transition probabilities. The probabilities may describe the model, but the profit vector records the profit for each specified outcome separately.
The convention assumes that the amounts are on a consistent annual accounting basis: investment income is included once, expenses and benefits are outgoes, and opening and closing reserves measure the insurer’s liability at the start and end of the year. A cash flow paid at a different time must be valued or accumulated consistently with the chosen basis before it is combined with the other amounts. If a benefit is already included in \(C\), do not include it a second time as a reserve movement.
A state-dependent profit vector lists amounts conditional on their respective state or transition outcomes. For example, if a policy starts active, its annual profit on the active-to-active outcome and its profit on the active-to-disabled outcome can be separate components. Each component uses the reserve appropriate to that outcome: the opening reserve for the starting state and the closing reserve for the end-of-year state or transition.
Why the Reserve Change Is Subtracted
The reserve is a liability, not an additional cash payment. The reserve adjustment reconciles the insurer’s end-of-year assets with the amount needed to support the closing liability. To see the sign, suppose opening assets equal the opening reserve. After premiums, investment income, expenses, and claims, the assets available at year-end are the opening assets plus net income and outgo. Profit is the excess of those assets over the closing reserve.
Thus, if the reserve increases by \$100, the insurer must retain \$100 more to support the liability, reducing current profit by \$100. If the reserve falls by \$100, that liability release adds \$100 to current profit. This reasoning depends on comparing reserves measured on a consistent basis and on treating the opening reserve as the assets available to support the policy at the start of the year.
Constructing a State-Dependent Profit Vector
Use positive magnitudes for premiums, expenses, claims, and reserves in the data table; apply the signs in the formula rather than trying to remember a separate sign for every product feature. Investment income is positive when earned and negative when it is an investment loss. A refund or benefit paid is an outflow and belongs with claims or other benefit payments.
Identify the starting state and the state or transition condition for the profit amount. Do not average over possible outcomes at this stage.
Record premium inflows and investment income as positive amounts; record expenses and benefits as positive outgo magnitudes to be subtracted.
Calculate closing reserve minus opening reserve. A positive result is an increase in liability; a negative result is a release.
Calculate \(P+I-E-C-(V_{\text{close}}-V_{\text{open}})\). Check that an increase in reserve reduces profit and a release increases it.
A terminating state often has a zero closing reserve if all obligations have been settled by year-end. That is not automatic: if an obligation remains after the event, the appropriate closing liability must still be included. Likewise, use the reserve specified for the actual destination state, not the reserve for a different outcome.
Worked Example: Active Policy Continuing in the Same State
A hypothetical one-year disability contract begins in the active state. Its opening reserve and opening supporting assets are \$900. A premium of \$300 is received at the start of the year and invested with those assets. The effective annual investment return is 4%, so the income on \$1,200 of invested assets is \$48. At year-end, expenses are \$100, no claim is paid, and the active-state closing reserve is \$1,050. For context, assume the one-year outcome distribution is active-to-active with probability 0.95, active-to-disabled with probability 0.04, and active-to-death with probability 0.01. Calculate the conditional profit for active-to-active.
Method. Apply the annual profit identity to the specified outcome only. The probabilities describe the model but do not enter this conditional calculation.
The reserve change is
The profit is therefore
Check. The reserve liability rises by \$150, so it reduces the \$248 net amount from premium, investment income, and expenses to \$98. The result is conditional on the active-to-active outcome; multiplying it by 0.95 would instead begin a probability-weighted calculation and is not part of this step.
Worked Example: Diagnosing an Incorrect Reserve Sign
Consider a hypothetical policy that starts active and ends the year disabled. Suppose the annual effective investment return is 3.2%, the opening assets and reserve are each \$1,000, and investment income is \$32. A \$240 premium is received at year-end; expenses are \$18; and the disability benefit paid is \$500. The closing reserve for the disabled state is \$850. Assume the outcome distribution from active is active with probability 0.90, disabled with probability 0.08, and death with probability 0.02. A student adds the reserve change to the cash-flow subtotal. Find the correct profit and the student’s result.
Method. Keep the reserve movement as closing minus opening, then subtract that change. First calculate the cash-flow subtotal:
Next calculate the reserve change:
The reserve has fallen by \$150, so the correct calculation subtracts the negative change:
The student instead adds the reserve change to the cash-flow subtotal:
Check. The \$150 reserve release should increase profit relative to the cash-flow subtotal of \(-\$246\), giving \(-\$96\), not \(-\$396\). The result remains a loss because the claims and expenses exceed the other amounts. The transition probabilities do not change either conditional calculation.
Worked Example: Two Outcomes and a Terminating-State Reserve
A hypothetical contract starts with opening assets and reserve of \$500. Its annual effective investment return is 4%, producing investment income of \$20. A \$500 premium is received at year-end, expenses are \$15, and the outcome is either active-to-active or active-to-death. Assume probabilities 0.99 and 0.01, respectively. No claim is paid if the policy remains active, and the active-state closing reserve is \$650. If death occurs, a \$800 death benefit is paid and all obligations are settled, so the closing reserve is zero. Construct the two conditional profit amounts.
Method. Use the same opening reserve, premium, investment income, and expenses for each outcome, but use each outcome’s own claim and closing reserve.
For active-to-active, the reserve change is \(650-500=150\), and no claim is paid:
For active-to-death, the reserve change is \(0-500=-500\), and the death benefit is \$800:
Check. In the active outcome, the \$150 reserve increase reduces profit. In the death outcome, the \$500 reserve release offsets part of the benefit outgo; it does not erase the \$800 claim. The conditional profit vector, in the order active-to-active and active-to-death, is \((\$355,\ \$205)\). No probability weighting has been applied.
Common Sign Errors and Their Checks
The most frequent error is to calculate \(V_{\text{close}}-V_{\text{open}}\) correctly and then add it to the cash-flow subtotal. The reserve movement is subtracted. An equivalent safe expression is \(+V_{\text{open}}-V_{\text{close}}\), which often makes the liability interpretation clearer.
| Situation | Reserve change | Effect on profit |
|---|---|---|
| Closing reserve exceeds opening reserve | Positive | Reduces profit |
| Closing reserve is below opening reserve | Negative | Increases profit |
| Closing reserve is zero after settlement | Negative if opening reserve is positive | Releases the opening reserve into current profit |
- Do not reverse the change. The defined change is closing reserve minus opening reserve, not the reverse.
- Do not give a claim a positive sign. When the data state a positive claim amount, subtract it as an insurer outgo.
- Do not use the wrong state’s reserve. The closing liability must correspond to the outcome being evaluated.
- Do not probability-weight a conditional component. Keep each outcome’s profit separate when constructing the vector.
- Do not double-count investment return. If investment income is already specified, include it once rather than also increasing a cash flow for the same return.
Check Your Understanding
For each question, calculate conditional profit using the stated annual convention. Unless a question asks otherwise, do not weight outcomes by their probabilities.
- A hypothetical policy has a \$400 premium, \$24 investment income, \$30 expenses, no claim, an opening reserve of \$200, and a closing reserve of \$260. Find annual profit. (Answer: \$334)
- A policy terminates on death. Its premium is \$150, investment income is \$12, expenses are \$7, the death benefit is \$300, and the reserve falls from \$250 to zero. Find profit. (Answer: \$105)
- A policy has premium \$80, investment income \$6, expenses \$10, and a benefit payment of \$20. Its reserve falls from \$400 to \$350. Find profit and state whether the reserve movement raises or lowers it. (Answer: \$106; the reserve release raises profit by \$50)
- A contract starts active with reserve \$500. Its premium is \$100, investment income is \$40, and expenses are \$10. On active-to-active, the claim is zero and the closing reserve is \$560. On active-to-exit, the benefit is \$200 and the closing reserve is zero. Give the two conditional profits in that order. (Answer: active-to-active \$70; active-to-exit \$430)
- A student uses premium \$240, investment income \$32, expenses \$18, benefit \$500, opening reserve \$1,000, and closing reserve \$850, but adds the reserve change to the cash-flow subtotal. State the correct profit and the student’s incorrect result. (Answer: Correct profit \(-\$96\); incorrect result \(-\$396\))