Modified Reserve Changes By Policy Year

Why Track a Modified Reserve Change?

A reserve at the start of a policy year and a reserve at the start of the next year are both measured for policies still in force at those respective times. The difference between them is not simply the interest earned on the opening reserve. Premiums and expenses enter during the year, benefits are paid on decrements, and the closing reserve applies only to survivors. A sound year-to-year calculation keeps each of these effects separate.

The calculation is made on a specified modified reserve basis. In particular, the premium, expense, benefit, interest, and decrement assumptions used in the recursion must be consistent with that basis. A modified premium or expense allowance is not interchangeable with a gross cash flow unless the reserve definition explicitly makes it so.

Definition: Consider one policy year, from time $t$ to time $t+1$. Let $V_t^{\mathrm{mod}}$ be the modified reserve at the start of the year per policy in force. Let $P_t$ be the premium received and $E_t$ the expense incurred at the start of the year. Let $i$ be the effective annual interest rate, ${}_1q_t$ the probability of the specified decrement during the year, and ${}_1p_t=1-{}_1q_t$ the probability of surviving in force to year-end. Let $B_{t+1}$ be the benefit paid at year-end if the decrement occurs. The modified reserve at the next year-end, conditional on survival, is $V_{t+1}^{\mathrm{mod}}$.

The One-Year Reserve Recursion

Assume the opening reserve, premium, and expense are available at the start of the year; the net amount earns interest at rate $i$ for the full year; the decrement benefit is paid at year-end; and $V_{t+1}^{\mathrm{mod}}$ is held at year-end for each survivor. For this one-decrement model, the expected year-end value of the cohort fund must equal the expected year-end outgo and the reserve held for survivors:

(V_t^{\mathrm{mod}}+P_t-E_t)(1+i) ={}_1q_t B_{t+1}+{}_1p_t V_{t+1}^{\mathrm{mod}}.

This is an expected-value reserve balance, not a claim that each individual policy has the same cash flow. The decrement group receives benefits, while the surviving group carries the closing reserve. Solving for that survivor reserve gives

V_{t+1}^{\mathrm{mod}} = \frac{(V_t^{\mathrm{mod}}+P_t-E_t)(1+i)-{}_1q_t B_{t+1}} {{}_1p_t}.

To obtain the change in reserve, subtract $V_t^{\mathrm{mod}}$ from both sides. Since ${}_1p_t=1-{}_1q_t$, the term involving the opening reserve simplifies as $(1+i)V_t^{\mathrm{mod}}-{}_1p_tV_t^{\mathrm{mod}}=(i+{}_1q_t)V_t^{\mathrm{mod}}$. Equivalently, the change can be written as

V_{t+1}^{\mathrm{mod}}-V_t^{\mathrm{mod}} = \frac{ iV_t^{\mathrm{mod}} +(1+i)(P_t-E_t) -{}_1q_t(B_{t+1}-V_t^{\mathrm{mod}}) }{{}_1p_t}.

The numerator separates the cash-flow contributions. The opening reserve contributes interest $iV_t^{\mathrm{mod}}$; the start-of-year premium and expense contribute their net amount accumulated for a year, $(1+i)(P_t-E_t)$; and the decrement term compares the benefit paid on a decrement with the opening reserve that would otherwise have supported the policy. The result is divided by the survival probability because the closing reserve is measured per survivor, not per policy initially in force.

Accumulate the available start-of-year amount.
Calculate $(V_t^{\mathrm{mod}}+P_t-E_t)(1+i)$. This includes one year of interest on the opening reserve and net start-of-year cash flow.

Subtract expected decrement benefits.
Subtract ${}_1q_tB_{t+1}$, the expected benefit outgo per policy initially in force.

Condition on survival.
Divide the remaining cohort amount by ${}_1p_t$ to obtain the reserve per survivor.

Worked Example: Separate Cohort Value From Survivor Reserve

A hypothetical annual contract has $V_t^{\mathrm{mod}}=\$300$, $P_t=\$40$, $E_t=\$10$, $i=0.02$, ${}_1q_t=0.05$, and $B_{t+1}=\$800$. The premium and expense are paid at the start of the year, and the decrement benefit is paid at year-end. Find the expected year-end amount remaining per policy initially in force, the next reserve per survivor, and the reserve change.

Method. Apply the cohort balance first, then divide by the survival probability. This avoids treating the expected cohort amount as though it were already a reserve per survivor.

The amount available at the start of the year is $300+40-10=\$330$. After interest, the year-end amount before decrement benefits is $330(1.02)=\$336.60$. The expected benefit outgo per initial policy is $0.05(800)=\$40.00$. Therefore, the expected amount remaining for survivors, still expressed per initial policy, is $336.60-40.00=\$296.60$.

The survival probability is $1-0.05=0.95$. Thus the reserve per survivor is $296.60/0.95=\$312.210526\ldots$, or \$312.21 to the nearest cent. The change from the opening reserve is $312.210526\ldots-300=\$12.210526\ldots$, or \$12.21.

Check. The survivor reserve weighted by the probability of survival is $0.95(312.210526\ldots)=\$296.60$. Adding the expected benefit outgo gives $296.60+40.00=\$336.60$, exactly the accumulated amount before benefits.

Interpreting the Components of the Change

The change formula is useful because it makes the reserve movement interpretable. For example, when $B_{t+1}$ exceeds $V_t^{\mathrm{mod}}$, a decrement removes more at year-end than the opening reserve associated with that policy. The term $-{}_1q_t(B_{t+1}-V_t^{\mathrm{mod}})$ therefore reduces the change numerator. Conversely, a benefit below the opening reserve makes that term positive.

This comparison is not a substitute for the full recursion. The survival divisor still matters: the reserve is calculated per survivor. Even if the cohort balance is correct, omitting that division generally gives the wrong closing reserve.

Worked Example: Calculate a Change Directly

For a hypothetical contract, $V_t^{\mathrm{mod}}=\$120$, $P_t=\$15$, $E_t=\$3$, $i=0.05$, ${}_1q_t=0.02$, and $B_{t+1}=\$400$. Cash flows have the timing stated in the definition. Find the reserve change and next reserve, to the nearest cent.

Method. Use the separated change formula, then add the change to the opening reserve. The common mistake to avoid is using the benefit amount $B_{t+1}$ without accounting for the opening reserve associated with the decrement in the change term.

First, ${}_1p_t=1-0.02=0.98$. The interest contribution is $iV_t^{\mathrm{mod}}=0.05(120)=\$6.00$. The accumulated net premium and expense contribution is $(1+i)(P_t-E_t)=1.05(15-3)=1.05(12)=\$12.60$. The decrement comparison is ${}_1q_t(B_{t+1}-V_t^{\mathrm{mod}})=0.02(400-120)=0.02(280)=\$5.60$.

The change is therefore $(6.00+12.60-5.60)/0.98=13.00/0.98=\$13.265306\ldots$, or \$13.27. The next reserve is $120+13.265306\ldots=\$133.265306\ldots$, or \$133.27.

Check. The accumulated start-of-year amount is $(120+15-3)(1.05)=132(1.05)=\$138.60$. Expected benefits are $0.02(400)=\$8.00$, leaving $\$130.60$ per initial policy for survivors. Dividing by $0.98$ gives $130.60/0.98=\$133.265306\ldots$, consistent with the result.

Worked Example: A Decrement Can Reduce the Reserve

A hypothetical policy has $V_t^{\mathrm{mod}}=\$250$, $P_t=\$20$, $E_t=\$8$, $i=0.04$, ${}_1q_t=0.10$, and $B_{t+1}=\$700$. Premium and expense are paid at the beginning of the year; the decrement benefit is paid at year-end. Find the next reserve per survivor and the change, to the nearest cent.

Method. Calculate the survivor reserve from the cohort balance and compare it with the opening reserve. This example highlights that a positive interest and net premium contribution do not guarantee that the reserve will increase.

The amount accumulated before benefits is $(250+20-8)(1.04)=262(1.04)=\$272.48$. The expected decrement benefit outgo per initial policy is $0.10(700)=\$70.00$. The amount remaining for survivors per initial policy is $272.48-70.00=\$202.48$. Since ${}_1p_t=0.90$, the next reserve is $202.48/0.90=\$224.977777\ldots$, or \$224.98.

The reserve change is $224.977777\ldots-250=-\$25.022222\ldots$, or -\$25.02. The change formula gives the same result: the numerator is $0.04(250)+1.04(20-8)-0.10(700-250)=10+12.48-45=-22.52$, and $-22.52/0.90=-\$25.022222\ldots$.

Check. Weighting the closing reserve by survival gives $0.90(224.977777\ldots)=\$202.48$. Adding the expected benefit outgo of $\$70.00$ recovers $\$272.48$, the accumulated amount before benefits.

Worked Example: Reconcile an Expected Cohort Amount

A hypothetical cohort has an opening reserve of $\$90$ per policy in force, a beginning-of-year premium of $\$12$, a beginning-of-year expense of $\$2$, and an annual effective interest rate of $3\%$. Expected end-of-year decrement benefit outgo is $\$25$ per policy initially in force, and survival probability is $0.95$. Find the expected amount remaining per initial policy and the reserve per survivor, to the nearest cent.

Method. Preserve the units of each amount: first calculate the cohort amount per initial policy, and only then condition on survival.

The start-of-year amount is $90+12-2=\$100$. Accumulating at $3\%$ gives $100(1.03)=\$103$. Subtracting the stated expected benefit outgo gives an expected amount remaining of $103-25=\$78.00$ per initial policy.

The reserve per survivor is $78/0.95=\$82.105263\ldots$, or \$82.11. The amount of $\$78.00$ is not itself the reserve per survivor; it is the survivor reserve weighted by the probability of survival.

Check. The survivor reserve weighted by survival is $0.95(82.105263\ldots)=\$78.00$. Adding the expected benefit outgo gives $78+25=\$103$, which matches the accumulated cohort amount.

Assumptions, Timing, and Common Traps

The displayed recursion applies to the stated annual timing: premium and expense at the start of the year, interest over the year, and decrement benefit and closing survivor reserve at year-end. If a premium or expense is paid at another time, its accumulation to year-end must change. If a benefit is paid at the moment of decrement rather than at year-end, its interest accumulation may depend on the decrement time; the simple term ${}_1q_tB_{t+1}$ is then not generally the correct expected year-end outgo. The recursion also presumes a single specified decrement probability and a single associated benefit amount. Where different decrement causes have different benefits, expected benefit outgo must reflect the cause-specific probabilities and benefits.

A frequent error is to calculate the cohort remainder and report it directly as $V_{t+1}^{\mathrm{mod}}$. For example, in the first worked example this error would report $\$296.60$, rather than $\$296.60/0.95=\$312.21$. Another error is to use $-{}_1q_tB_{t+1}$ as the decrement term in the change formula without also accounting for the opening reserve removed when policies decrement. The correct change term is $-{}_1q_t(B_{t+1}-V_t^{\mathrm{mod}})$, together with division by ${}_1p_t$.

Key takeaway: With beginning-of-year premium and expense, year-end decrement benefits, and one decrement, first form the expected cohort balance and then divide by survival probability. For the change directly, separate interest, accumulated net premium and expense, and the decrement comparison $B_{t+1}-V_t^{\mathrm{mod}}$; retain the survival divisor.

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. On an exam-style problem, state the timing and basis, identify whether an amount is per initial policy or per survivor, and verify the answer by substituting it back into the cohort balance.

Check Your Understanding

For each hypothetical annual contract, use beginning-of-year premium and expense, year-end decrement benefits, and the one-decrement recursion. Round monetary answers to the nearest cent.

A policy has $V_t^{\mathrm{mod}}=\$200$, $P_t=\$30$, $E_t=\$5$, $i=0.03$, ${}_1q_t=0.04$, and $B_{t+1}=\$500$. Find the expected amount remaining per initial policy, the next reserve per survivor, and the reserve change. (Answer: expected cohort amount $\$211.75$; next reserve $\$220.57$; change $+\$20.57$)
A policy has $V_t^{\mathrm{mod}}=\$80$, $P_t=\$12$, $E_t=\$2$, $i=0.04$, ${}_1q_t=0.05$, and $B_{t+1}=\$300$. Find the next reserve and reserve change. (Answer: next reserve $\$82.74$; change $+\$2.74$)
A policy has $V_t^{\mathrm{mod}}=\$450$, $P_t=\$25$, $E_t=\$9$, $i=0.02$, ${}_1q_t=0.03$, and $B_{t+1}=\$900$. Find the next reserve and reserve change. (Answer: next reserve $\$462.19$; change $+\$12.19$)
A policy has $V_t^{\mathrm{mod}}=\$160$, $P_t=\$18$, $E_t=\$4$, $i=0.05$, and ${}_1q_t=0.02$. Its next reserve per survivor is $\$175$. Find the year-end decrement benefit $B_{t+1}$. (Answer: $\$560.00$)
A hypothetical policy has $V_0^{\mathrm{mod}}=\$100$. In year 0, $P_0=\$20$, $E_0=\$5$, $i=0.04$, ${}_1q_0=0.10$, and $B_1=\$500$. In year 1, use $P_1=\$10$, $E_1=\$3$, $i=0.03$, ${}_1q_1=0.05$, and $B_2=\$400$. Calculate the reserve at the start of year 2 and its change from the start of year 1. (Answer: $V_2^{\mathrm{mod}}=\$70.38$; year-1 change $-\$6.95$)

What You'll Learn