Discrete Reserve Recursions for Annual Profit Projections

Why a Discrete Reserve Recursion Matters

Annual profit projections track how premiums, expenses, benefits and reserves change from one policy year to the next. In a multi-state contract, those cash flows can depend on the state at the start of the year and on the transition during the year. A reserve recursion brings these elements together: it gives the opening reserve needed, on a specified basis, to support the expected end-of-year benefit payments and closing reserves.

The calculation is particularly sensitive to timing. A premium received at the start of the year earns interest for the full year; a benefit paid at the end does not. A benefit or closing reserve must also be weighted by the probability of reaching the state in which it is payable. The conventions below make those points explicit. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Model and Timing Conventions

Definition: Let $j$ denote the state at the start of policy year $t$, and let $k$ denote the state at the end of that year. Write $p_t^{jk}$ for the probability of transition from $j$ to $k$ during the year, conditional on starting in $j$. The probabilities across all possible end states, including any terminating state, sum to 1. Let $V_t^j$ be the reserve at the start of the year, before that year's premium and start-of-year expense. Let $P_t^j$ and $E_t^j$ be the premium received and expense paid at the start of the year. Let $B_t^{jk}$ be the benefit paid at year-end if the transition is from $j$ to $k$, and let $V_{t+1}^k$ be the reserve immediately after year-end cash flows. A terminating state has closing reserve 0.

Assume an effective annual interest rate $i$, with no cash flows between the stated start- and end-of-year times. Premiums and expenses depend on the known starting state; benefits may depend on the transition. The reserve is calculated on the same transition, interest and cash-flow assumptions as the projection. Other timing conventions are possible, but they require changing the accumulation or discount factors consistently.

At the start of the year, the funds available after the premium and expense are $V_t^j+P_t^j-E_t^j$. They accumulate for one year. At year-end, the insurer pays the benefit associated with the transition and sets aside the reserve for the resulting state. Thus the expected end-of-year amount required is the sum of each possible benefit plus closing reserve, weighted by its transition probability.

The One-Year Recursion and Its Justification

\left(V_t^j+P_t^j-E_t^j\right)(1+i) = \sum_k p_t^{jk}\left(B_t^{jk}+V_{t+1}^k\right).

Solving for the opening reserve gives the discrete reserve recursion:

V_t^j = -P_t^j+E_t^j + \frac{\displaystyle\sum_k p_t^{jk}\left(B_t^{jk}+V_{t+1}^k\right)}{1+i}.

The justification is a one-year balance of funds on the reserve basis. Conditional on starting in state $j$, the end-of-year requirement is $B_t^{jk}+V_{t+1}^k$ if the contract moves to state $k$. Its conditional expectation is therefore $\sum_k p_t^{jk}(B_t^{jk}+V_{t+1}^k)$. Start-of-year funds earn interest at rate $i$, so their accumulated value must equal that expected requirement. Dividing by $1+i$ discounts the requirement to the beginning of the year, and rearranging accounts for the premium and expense.

This equation is an expected-value reserve recursion. It does not say that the insurer's actual profit is zero for every transition. Outcomes can differ substantially by state; zero expected profit follows only when the reserve basis and the projection assumptions agree.

Worked Example: Active or Dead at Year-End

A hypothetical one-year contract starts in the active state. The annual effective interest rate is $5\%$. At the start of the year, the insurer receives a premium of $100 and pays an expense of $10. The policyholder remains active with probability $0.90$, or dies with probability $0.10$. At year-end the death benefit is $500 if death occurs; there is no active-state benefit. The closing active reserve is $200 and the closing death-state reserve is 0. Find the opening active-state reserve and the profit under each outcome.

Method. Apply the recursion to the two possible end states. The probability-weighted year-end requirement is $0.90(0+200)+0.10(500+0)=180+50=230$.

The net start-of-year cash flow excluding reserve is $P-E=100-10=90$. Therefore,

V_0^{A} = -100+10+\frac{230}{1.05} = -90+219.047619\ldots = \$129.05.

For a direct check, before rounding, the accumulated opening funds are $(129.047619\ldots+100-10)(1.05)=230$, equal to the probability-weighted year-end requirement.

Define realized year-end profit as accumulated opening reserve and net start-of-year cash flow, less the benefit and closing reserve for the observed outcome. The accumulated amount is $230. If the policyholder remains active, profit is $230-0-200=\$30$. If death occurs, profit is $230-500-0=-\$270$. The expected profit is $0.90(30)+0.10(-270)=27-27=\$0$, as required by the reserve basis.

Common mistake. Treating death as certain would give $-90+500/1.05=\$386.19$, which is not the opening reserve for this model. The death benefit is payable only with probability $0.10$, and the active closing reserve is also probability-weighted. Both possible transitions must be included.

State-Dependent Cash Flows

The same recursion applies when premiums, expenses and benefits vary by state. A premium payable only while active should be included in $P_t^j$ when the starting state is active and omitted in a different starting state. A disability income payment may be represented as a year-end benefit on transitions that end in disability, or as a state-contingent annual benefit if that matches the product's specified timing. The model must not count a payment both as a transition benefit and as a separate state benefit.

When calculating a reserve for one starting state, include every possible end state with its probability, benefit and closing reserve. A terminating state contributes any benefit due at termination but has no later reserve. For a non-terminating state, its closing reserve is part of the year-end requirement even if no benefit is paid on that transition.

Worked Example: Disability Benefits and a Closing Reserve

A hypothetical income-protection contract starts in the active state. The annual effective interest rate is $4\%$. At the start of the year it receives a premium of $200 and pays an expense of $20. The end-of-year state is active with probability $0.80$ or disabled with probability $0.20$. There is no benefit if active; a $300 benefit is paid if disabled. Closing reserves are $120 in the active state and $400 in the disabled state. Calculate the opening active reserve and check the expected profit.

Method. First calculate the year-end requirement in each end state. It is $0+120=120$ if active and $300+400=700$ if disabled. The expected requirement is $0.80(120)+0.20(700)=96+140=236$.

The net start-of-year cash flow is $200-20=180$. Applying the recursion,

V_0^{A} = -200+20+\frac{236}{1.04} = -180+226.923076\ldots = \$46.92.

Before rounding, accumulated opening funds are $(46.923076\ldots+200-20)(1.04)=236$. If the end state is active, realized profit is $236-120=\$116$. If disabled, it is $236-300-400=-\$464$. The expected profit is $0.80(116)+0.20(-464)=92.80-92.80=\$0$.

A tempting but incorrect calculation is to add the disabled benefit and reserve as though disability were certain, then discount: $-180+700/1.04=\$493.08$. The correction is to weight each possible end-state amount by its transition probability before discounting. This example also shows why the reserve is not simply the discounted closing reserve: the transition-contingent benefit must be included.

Calculating Reserves Across Several States

For a multi-state contract, calculate a recursion for each non-terminating starting state. The transition probabilities in a row belong to that starting state; they must not be reused for another state unless the model specifies identical transition probabilities. The resulting reserves are state-specific and can be used as closing reserves in the next year's recursion.

Worked Example: Disability, Recovery and Death

A hypothetical contract has two non-terminating states: disabled ($D$) and recovered ($R$), plus a terminating death state. The annual effective interest rate is $5\%$. At the start of the year, a disabled policyholder pays no premium and incurs a $20 expense; a recovered policyholder pays a $150 premium and incurs a $10 expense. From $D$, the end state is $D$ with probability $0.60$, $R$ with probability $0.30$, and death with probability $0.10$. From $R$, it is $R$ with probability $0.80$, $D$ with probability $0.10$, and death with probability $0.10$. At year-end, a $100 benefit is paid if the end state is $D$, and a $1,000 benefit if death occurs; no benefit is paid if the end state is $R$. Closing reserves are $V_{t+1}^{D}=\$500$, $V_{t+1}^{R}=\$200$, and 0 on death. Calculate both opening reserves and verify the profit expectation in each starting state.

Method for a disabled starting state. The year-end requirements are $600 for transition $D$ to $D$, $200 for $D$ to $R$, and $1,000 for death. Their probability-weighted total is $0.60(600)+0.30(200)+0.10(1{,}000)=360+60+100=520$. The starting-state premium is 0 and expense is $20, so

V_t^D = -0+20+\frac{520}{1.05} = 20+495.238095\ldots = \$515.24.

The accumulated opening funds before rounding are $(515.238095\ldots-20)(1.05)=520$. Realized profits for transitions to $D$, $R$, and death are, respectively, $520-600=-\$80$, $520-200=\$320$, and $520-1{,}000=-\$480$. Their expectation is $0.60(-80)+0.30(320)+0.10(-480)=-48+96-48=\$0$.

Method for a recovered starting state. The year-end requirements are $200 for transition $R$ to $R$, $600 for $R$ to $D$, and $1,000 for death. The weighted total is $0.80(200)+0.10(600)+0.10(1{,}000)=160+60+100=320$. The net start-of-year cash flow is $150-10=140$. Hence

V_t^R = -150+10+\frac{320}{1.05} = -140+304.761905\ldots = \$164.76.

The accumulated opening funds are $(164.761905\ldots+150-10)(1.05)=320$. Realized profits for transitions to $R$, $D$, and death are $320-200=\$120$, $320-600=-\$280$, and $320-1{,}000=-\$680$. Their expected value is $0.80(120)+0.10(-280)+0.10(-680)=96-28-68=\$0$. All reserves are rounded to cents only at the final step.

Profit Interpretation and Practical Checks

For a realized transition from $j$ to $k$, the one-year profit on this timing convention is

\pi_t^{jk} = \left(V_t^j+P_t^j-E_t^j\right)(1+i) - B_t^{jk} - V_{t+1}^k.

The reserve recursion implies that the conditional expected profit given starting state $j$ is zero when the transition probabilities and all cash-flow and interest assumptions are the ones used to calculate the reserve:

\sum_k p_t^{jk}\pi_t^{jk} = \left(V_t^j+P_t^j-E_t^j\right)(1+i) - \sum_k p_t^{jk}\left(B_t^{jk}+V_{t+1}^k\right) = 0.

This is a useful consistency check, not a claim about actual experience. If actual transition rates, interest earned or expenses differ from the valuation basis, the realized or projected profit may not be zero. The recursion still clarifies how the reserve, cash flows and closing obligations combine.

A reliable calculation follows four checks:

Fix the timing.
State which cash flows occur at the start and end of the year. Accumulate start-of-year net funds for a full year under the stated annual effective rate.

List all possible end states.
Check that the transition probabilities conditional on the starting state sum to 1, including terminating states.

Build the year-end requirement.
For each transition, add its year-end benefit to the closing reserve in the destination state, using zero reserve for a terminating state.

Reconcile the balance.
Check that accumulated opening funds equal the weighted year-end requirement, then use the same amounts to calculate each realized transition profit.

Common errors include omitting a closing reserve in a non-terminating state, applying the wrong transition probabilities, treating an end-of-year benefit as if it earned interest for the year, or applying interest to a premium that is actually received at year-end. If a contract uses a different payment timing, write the cash-flow balance for that timing rather than inserting the standard recursion unchanged.

Key takeaway: For each starting state, discount the probability-weighted sum of transition benefits and destination-state reserves, then adjust for start-of-year premiums and expenses. The resulting reserve makes expected one-year profit zero on the same assumptions, while individual transition profits may be positive or negative.

Check Your Understanding

For each problem, use the stated annual effective interest rate and timing conventions. Round money to cents only at the final step.

A contract starts active. The annual effective interest rate is $4\%$; a $80 premium and $5 expense occur at the start of the year. The end state is active with probability $0.75$ and disabled with probability $0.25$. There is no active benefit; the disabled benefit is $200. Closing reserves are $100 active and $300 disabled. Find the opening active reserve. (Answer: $117.31)
A contract starts disabled. The annual effective interest rate is $5\%$; there is no premium and a $12 start-of-year expense. The end state is disabled with probability $0.50$, recovered with probability $0.40$, or death with probability $0.10$. The year-end benefit is $80 in the disabled state, $0 in recovery and $600 on death. Closing reserves are $250 disabled, $100 recovered and 0 on death. Find the opening disabled reserve. (Answer: $264.38)
A projection uses a $90 opening reserve, a $120 start-of-year premium and a $10 start-of-year expense at $5\%$ interest. The end state is active with probability $0.80$, with a $120 closing reserve and no benefit, or death with probability $0.20$, with a $400 benefit and no closing reserve. Calculate the expected one-year profit under this projection. (Answer: $34.00)
A contract starts disabled. The annual effective interest rate is $4\%$; a $40 premium and an $8 expense occur at the start of the year. The end state is disabled with probability $0.70$, recovered with probability $0.20$, or death with probability $0.10$. The year-end benefit is $50 if disabled, $0 if recovered and $800 on death. Closing reserves are $300 disabled, $120 recovered and 0 on death. Find the opening disabled reserve. (Answer: $303.58)

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