Worked Three-Year Reserve-Adjusted Projection

Purpose and Projection Convention

A reserve-adjusted projection shows more than whether premiums collected in a year exceed claims and expenses. It also records the investment return earned on funds held during the year and the amount that must remain in reserve for policies still in force at year-end. The resulting annual profit can therefore be negative even when current-year premiums exceed current-year claims and expenses.

This tutorial develops a three-year projection for a hypothetical level-premium policy. The calculations use expected policy counts, so fractional in-force counts are appropriate: they represent expected numbers of policies, not literal partial policies. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Projection assumptions and timing: At the start of each year, every policy then in force pays its annual premium and has an opening reserve. Both amounts earn the effective annual investment rate $i$ for the full year. Deaths occur during the year; the conditional probability of death is $q_t$, and the probability of surviving is $p_t=1-q_t$. Death benefits and expenses are paid at year-end and earn no investment income in the projection. At year-end, the reserve $V_t$ is set aside for each surviving policy. There are no lapses, new policies, or other cash flows. For a three-year term, the terminal reserve is $V_3=0$.

Let $N_{t-1}$ be the expected number of policies in force at the start of year $t$, and let $N_t$ be the expected number surviving to its end. With $N_0=1$ for a projection per original policy, $N_t=N_{t-1}p_t$. Let $P_t$ be the premium per opening policy, $B_t$ the death benefit, and $e_t$ the expense per opening policy. The annual premium is level when $P_t=P$ in each year.

For this convention, annual profit has two useful expressions. Profit per policy in force at the start of the year is $\pi_t$; total expected profit for the projected cohort is $\Pi_t$.

\pi_t=(V_{t-1}+P_t)(1+i)-q_tB_t-e_t-p_tV_t, \qquad \Pi_t=N_{t-1}\pi_t.

The term $p_tV_t$ is the expected year-end reserve allocation per opening policy: only the proportion $p_t$ survives and receives a reserve. If $V_t$ is a reserve per surviving policy, it is not itself the reserve transfer per opening policy.

Why the Reserve-Adjusted Formula Holds

The formula follows by accounting for the funds attributable to one opening policy. At the start of the year, the opening reserve and premium are $V_{t-1}+P_t$. They accumulate to $(V_{t-1}+P_t)(1+i)$. Expected year-end claim outgo per opening policy is $q_tB_t$, and the expense outgo is $e_t$. After paying those amounts, the fund must retain $V_t$ for each survivor. The expected amount retained per opening policy is $p_tV_t$. The balance is profit.

\begin{aligned} \text{accumulated opening funds} &-\text{expected claims} -\text{expenses} -\text{expected closing reserve}\\ &=(V_{t-1}+P_t)(1+i)-q_tB_t-e_t-p_tV_t. \end{aligned}

Multiplying by $N_{t-1}$ converts the per-opening-policy result into the expected cohort result. This gives $\Pi_t=N_{t-1}\pi_t$. In particular, multiplying by the opening in-force count is necessary in later years; a per-policy result is not yet the total profit for the original cohort.

The reserve schedule is an input to this projection. It may have been calculated on a valuation basis that differs from the projection assumptions, so the method does not imply that projected profit must be zero. A reserve increase can absorb current funds, while a reserve decrease can release funds and increase current profit.

Worked Three-Year Projection

The following examples apply the same accounting method to different hypothetical contracts. In each, the mortality assumption is a year-specific Bernoulli death event conditional on being in force at the start of that year, with $P(\text{death in year }t)=q_t$ and $P(\text{survival})=p_t$. These are annual probabilities, not forces of mortality.

Worked Example: Projecting Profit for a Three-Year Term Policy

A hypothetical three-year term policy charges a level premium of $120 at the start of each year in which it is in force. The death benefit is $1,000, and the year-end expense is $10 per opening policy. The effective annual investment rate is $i=5\%$. In each year, the conditional death probability is $q_t=0.10$, so $p_t=0.90$. The reserve per surviving policy is $V_0=0$, $V_1=40$, $V_2=20$, and $V_3=0$. Project each year’s profit per opening policy and for the original cohort.

Method. First project expected in-force counts. Then calculate the accumulated opening reserve and premium, subtract expected claims and expenses, and fund reserves for expected survivors.

The expected counts are $N_0=1$, $N_1=1(0.90)=0.90$, and $N_2=0.90(0.90)=0.81$. Thus the opening counts in years 1, 2, and 3 are $1$, $0.90$, and $0.81$, respectively.

Year 1. Accumulated opening funds are $(0+120)(1.05)=126$. Expected claims are $0.10(1{,}000)=100$, expenses are 10, and expected closing reserves are $0.90(40)=36$. Therefore $\pi_1=126-100-10-36=-20$, and $\Pi_1=1(-20)=-20$.

Year 2. Accumulated opening funds per opening policy are $(40+120)(1.05)=168$. Expected claims are 100, expenses are 10, and expected closing reserves are $0.90(20)=18$. Therefore $\pi_2=168-100-10-18=40$. The cohort profit is $\Pi_2=0.90(40)=36$.

Year 3. Accumulated opening funds per opening policy are $(20+120)(1.05)=147$. Expected claims are 100, expenses are 10, and the terminal reserve is zero. Therefore $\pi_3=147-100-10-0=37$. The cohort profit is $\Pi_3=0.81(37)=29.97$.

The three cohort profits are $-20$, $36$, and $29.97$. Their sum is $45.97$. The check is that each annual cohort result equals its per-opening-policy result times that year’s opening count: $-20(1)=-20$, $40(0.90)=36$, and $37(0.81)=29.97$.

Common mistake avoided: The year 2 reserve transfer is $0.90(20)=18$ per opening policy, not 20. The closing reserve is held only for the policies that survive the year.

Worked Example: Projecting Changing Mortality Through the Term

A hypothetical three-year policy has a level premium of $90, a death benefit of $500, and a year-end expense of $8 per opening policy. The effective annual investment rate is $4\%$. Conditional death probabilities are $q_1=0.05$, $q_2=0.10$, and $q_3=0.20$, with survival probabilities $0.95$, $0.90$, and $0.80$. The reserve schedule is $V_0=0$, $V_1=25$, $V_2=15$, and $V_3=0$. Calculate expected annual cohort profits.

Method. Apply the conditional survival probability year by year, then calculate annual profit per opening policy before multiplying by the appropriate expected opening count.

The expected opening counts are $N_0=1$, $N_1=1(0.95)=0.95$, and $N_2=0.95(0.90)=0.855$.

Year 1. Accumulated funds are $(0+90)(1.04)=93.60$. Expected claims are $0.05(500)=25$, expenses are 8, and expected closing reserves are $0.95(25)=23.75$. Hence $\pi_1=93.60-25-8-23.75=36.85$, and $\Pi_1=1(36.85)=36.85$.

Year 2. Accumulated funds per opening policy are $(25+90)(1.04)=119.60$. Expected claims are $0.10(500)=50$, expenses are 8, and expected closing reserves are $0.90(15)=13.50$. Hence $\pi_2=119.60-50-8-13.50=48.10$, and $\Pi_2=0.95(48.10)=45.695$.

Year 3. Accumulated funds per opening policy are $(15+90)(1.04)=109.20$. Expected claims are $0.20(500)=100$, expenses are 8, and the terminal reserve is zero. Hence $\pi_3=109.20-100-8=1.20$, and $\Pi_3=0.855(1.20)=1.026$.

The expected cohort profits, before rounding, are $36.85, $45.695, and $1.026. For reporting to the nearest cent, they are $36.85, $45.70, and $1.03. The in-force check for the final year is $0.95(0.90)=0.855$; using $0.90$ alone would omit the year 1 survival requirement.

Common mistake avoided: The year 3 opening count is the product of survival probabilities in both earlier years. Applying only the year 3 survival probability to the original policy would give the wrong cohort exposure.

Worked Example: Recognizing Profit Suppressed by a Reserve Increase

A hypothetical three-year policy has a level premium of $100, a death benefit of $600, and a year-end expense of $5 per opening policy. The effective annual investment rate is $5\%$. The conditional death probability is $0.10$ in each year, and the reserve schedule is $V_0=0$, $V_1=20$, $V_2=80$, and $V_3=0$. Calculate annual cohort profits and explain the year 2 result.

Method. Use the reserve-adjusted profit formula without changing the timing convention. In particular, include the accumulated opening reserve in funds and the expected closing reserve as an outgo.

Expected opening counts are $N_0=1$, $N_1=0.90$, and $N_2=0.90(0.90)=0.81$.

Year 1. Accumulated funds are $(0+100)(1.05)=105$. Expected claims are $0.10(600)=60$, expenses are 5, and expected closing reserves are $0.90(20)=18$. Thus $\pi_1=105-60-5-18=22$, and $\Pi_1=1(22)=22$.

Year 2. Accumulated funds per opening policy are $(20+100)(1.05)=126$. Expected claims are 60, expenses are 5, and expected closing reserves are $0.90(80)=72$. Thus $\pi_2=126-60-5-72=-11$, and $\Pi_2=0.90(-11)=-9.90$.

Year 3. Accumulated funds per opening policy are $(80+100)(1.05)=189$. Expected claims are 60, expenses are 5, and the terminal reserve is zero. Thus $\pi_3=189-60-5=124$, and $\Pi_3=0.81(124)=100.44$.

Year 2 has negative profit because the reserve rises from $20 to $80 per policy, requiring a large closing reserve allocation. The result is not evidence of an arithmetic error: year 3 begins with that larger reserve, earns investment income on it, and has no terminal reserve to fund. The cohort-profit check is $22-9.90+100.44=112.54$.

Common mistake avoided: Treating the $80 closing reserve as an expense on every opening policy would deduct $80 instead of the correct expected amount $0.90(80)=72$. Reserve funding is conditional on survival.

Checks and Common Projection Errors

A reliable projection keeps the per-policy and cohort calculations separate. The per-policy profit is conditional on being in force at the start of the year; the cohort profit includes the probability-weighted number of policies reaching that year. The following checks help identify inconsistencies before interpreting the result.

Check exposure: verify $N_t=N_{t-1}p_t$ for every year.
Check reserve units: $V_t$ is per surviving policy, while the expected reserve transfer per opening policy is $p_tV_t$.
Check investment timing: under the stated convention, opening reserve and beginning-of-year premium earn a full year of interest; year-end claims and expenses do not.
Check the terminal condition: if coverage ends after year 3 and no benefit remains payable to survivors, use $V_3=0$.
Check the projection basis: a reserve is not automatically consistent with the projection assumptions. A nonzero expected profit can result from basis differences or from the reserve schedule itself.

A frequent timing error is to subtract a closing reserve before accounting for the funds available at the start of the year, or to subtract the full reserve for every opening policy. The formula avoids both errors by accumulating opening funds first and then deducting the expected reserve allocated to survivors.

Check Your Understanding

Use the stated annual timing convention. Treat all mortality probabilities as conditional on being in force at the start of the year.

A one-year projection has $i=4\%$, $V_0=10$, premium $80, death probability $0.05$, death benefit $400, year-end expense $6, and $V_1=20$. Find profit per opening policy. (Answer: $48.60)
A three-year policy has $i=3\%$, level premium $70, benefit $200, year-end expense $4, and $q_t=0.20$ each year. Reserves are $V_0=0$, $V_1=15$, $V_2=10$, and $V_3=0$. Find year 2 cohort profit. (Answer: $28.44)
At the start of a year, 0.72 policies are expected to be in force. The survival probability is $0.85$, and the reserve per survivor at year-end is $50. What is the expected closing reserve allocation for the cohort? (Answer: $30.60)
For a one-year projection, $i=5\%$, $V_0=30$, premium $100, death probability $0.10$, benefit $500, and expense $5. What closing reserve per survivor makes expected profit per opening policy zero? (Answer: $90.56)
A three-year policy has conditional survival probabilities $0.90$, $0.80$, and $0.75$. Starting with one policy, how many are expected to be in force at the start of year 3? (Answer: 0.72)

What You'll Learn