Forward and Backward Reserve Calculations

Why Calculate Reserves in Both Directions?

A reserve sequence describes the policy liability at successive policy anniversaries. In a forward calculation, an initial reserve and the policy cash flows determine later reserves. In a backward calculation, a known later reserve and the intervening cash flows determine earlier reserves. These are two rearrangements of the same annual balance equation, not separate reserve methods.

The direction matters in practical calculations. A problem may give the reserve at issue and ask for a later value, or prescribe a reserve at a future duration and ask what an earlier value must have been. The arithmetic is usually straightforward once the timing convention is fixed. The main risk is mixing a reserve measured before a premium with one measured after it, or applying interest to cash flows at the wrong time.

Definition: Let ${}_kV_x$ denote the reserve at time $k$, immediately before the premium due at that time, for a policy issued at age $x$ and in force at time $k$. In the annual model used here, a premium $P_k$ is paid at time $k$, and an expense $e_k$, if any, is also paid then. At the end of the year, a death benefit $B_{k+1}$ is paid if death occurs during that year. If the policyholder survives, the reserve at time $k+1$ is ${}_{k+1}V_x$, immediately before the next premium.

Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Write $q_{x+k}$ for the probability of death during the year from age $x+k$, and $p_{x+k}=1-q_{x+k}$ for the probability of survival. The mortality probabilities are known inputs to the calculation. This setup assumes annual premiums and expenses are paid at the start of the year, while death benefits and the next reserve are measured at the end of the year. Other payment timings require a corresponding change to the equation.

Theorem: Under this timing convention, the annual reserve equation is

\left({}_kV_x+P_k-e_k\right)(1+i) = q_{x+k}B_{k+1} + p_{x+k}{}_{k+1}V_x.

The left side is the amount available at the end of the year after accumulating the time-$k$ reserve, premium, and expense. The right side is the expected end-of-year amount required for the death claim or the surviving policy.

Why the Recursion Holds

Immediately after the time-$k$ premium and expense, the amount available is ${}_kV_x+P_k-e_k$. Accumulating this amount for one year gives the left side of the reserve equation. At the year end, death has probability $q_{x+k}$ and requires payment $B_{k+1}$; survival has probability $p_{x+k}$ and carries reserve ${}_{k+1}V_x$. The probability-weighted end-of-year requirement is therefore the right side. Equating available assets and expected requirements gives the recursion.

This is a conditional balance for a policy in force at time $k$. It does not say that the insurer will pay an expected fraction of a claim on an individual policy. The actual outcome is a death benefit or a surviving policy reserve; the probability-weighted expression is used to set the reserve on the stated assumptions.

Set the time convention.
Identify whether each reserve is measured before or after the premium, and place expenses and benefits at their stated payment times.

Write the year’s balance.
Substitute the premium, expense, interest rate, death probability, benefit, and the known reserve into the annual equation.

Solve in the required direction.
Isolate the later reserve for a forward calculation, or the earlier reserve for a backward calculation.

Check the result.
Substitute the calculated value back into the original balance equation and verify that both sides agree.

Forward and Backward Forms

Solving the balance equation for the later reserve gives the forward form. It requires $p_{x+k}>0$, because a reserve conditional on survival cannot be recovered from an equation that assigns zero probability to survival.

{}_{k+1}V_x = \frac{\left({}_kV_x+P_k-e_k\right)(1+i)-q_{x+k}B_{k+1}} {p_{x+k}}.

To calculate backward, first discount the expected end-of-year requirement, then remove the net amount contributed at time $k$. This rearrangement does not require division by $p_{x+k}$.

{}_kV_x = \frac{q_{x+k}B_{k+1}+p_{x+k}{}_{k+1}V_x}{1+i} -P_k+e_k.

For a net premium recursion, set expenses to zero. For a gross recursion, include the expenses at the times they are paid. In either case, $P_k-e_k$ is the net amount added at time $k$ under the convention above. Do not subtract an expense again if the stated premium or reserve has already been defined net of that expense.

Worked Example: Calculate Two Reserves Forward

A hypothetical annual life insurance policy has a reserve of ${}_0V_x=\$100$ just before the first premium. The annual premium is $\$100$, payable at the start of each year, with no expenses. The interest rate is $i=10\%$, and the death benefit of $\$500$ is paid at each year end on death. For the first year, $q_x=0.20$; for the second year, $q_{x+1}=0.10$. Calculate ${}_1V_x$ and ${}_2V_x$.

Method. Apply the forward recursion once for each year. The corresponding survival probabilities are $p_x=1-0.20=0.80$ and $p_{x+1}=1-0.10=0.90$.

First year. The reserve and premium available at time 0 are $\$100+\$100=\$200$. Accumulated for one year, this is $\$200(1.10)=\$220$. The probability-weighted death benefit is $0.20(\$500)=\$100$. Thus the amount assigned to survival is $\$220-\$100=\$120$, so

{}_1V_x=\frac{\$120}{0.80}=\$150.

Second year. At time 1, the reserve plus premium is $\$150+\$100=\$250$. Accumulation gives $\$250(1.10)=\$275$. The expected death-benefit amount is $0.10(\$500)=\$50$, leaving $\$275-\$50=\$225$ for survival. Therefore,

{}_2V_x=\frac{\$225}{0.90}=\$250.

Check. In year 1, $0.20(\$500)+0.80(\$150)=\$100+\$120=\$220$, matching the accumulated amount. In year 2, $0.10(\$500)+0.90(\$250)=\$50+\$225=\$275$. The sequence satisfies both annual balances. The common mistake this method avoids is subtracting the full death benefit rather than its probability-weighted amount.

Worked Example: Recover Earlier Reserves Backward

A hypothetical policy has an annual premium of $\$100$, no expenses, a death benefit of $\$500$ paid at each year end, and an effective annual interest rate of $10\%$. The one-year death probabilities are $q_x=0.20$ and $q_{x+1}=0.10$. If the reserve at time 2 is ${}_2V_x=\$250$, calculate ${}_1V_x$ and ${}_0V_x$.

Method. Use the backward form, starting with the latest known reserve. The survival probabilities are $0.80$ for the first year and $0.90$ for the second year.

From time 2 to time 1. The expected time-2 requirement is $0.10(\$500)+0.90(\$250)=\$50+\$225=\$275$. Discounting one year gives $\$275/1.10=\$250$. Subtract the time-1 premium of $\$100$:

{}_1V_x=\$250-\$100=\$150.

From time 1 to time 0. The expected time-1 requirement is $0.20(\$500)+0.80(\$150)=\$100+\$120=\$220$. Discounting gives $\$220/1.10=\$200$. Subtract the time-0 premium:

{}_0V_x=\$200-\$100=\$100.

Check. The reserves $\$100,\$150,\$250$ reproduce the forward balances from the previous year-by-year equations. The common mistake avoided here is subtracting the premium before discounting the end-of-year requirement. The premium is paid at the start of the year, so it is removed after that requirement has been discounted to the start of the year.

Varying Premiums and Expenses

The recursion remains the same when premiums or expenses vary by duration; use the values applying at each time $k$. A useful way to prevent sign errors is to combine the time-$k$ cash flows as $P_k-e_k$ before accumulating forward, or as $-P_k+e_k$ after discounting backward. An expense reduces the amount available to support future benefits, while a premium increases it.

Worked Example: Use Varying Cash Flows and Check Both Directions

A hypothetical policy has $i=10\%$, a $\$500$ death benefit paid at the end of either policy year, and no other benefit. In year 1, $q_x=0.20$, the premium is $P_0=\$140$, and the expense is $e_0=\$20$. In year 2, $q_{x+1}=0.10$, the premium is $P_1=\$115$, and the expense is $e_1=\$15$. Given ${}_0V_x=\$80$, calculate ${}_1V_x$ and ${}_2V_x$, then verify by calculating backward from ${}_2V_x$.

Method. Apply the forward equation using the net time-zero cash flow $P_0-e_0=\$120$ and the net time-one cash flow $P_1-e_1=\$100$.

First year. The amount accumulated to time 1 is $(\$80+\$140-\$20)(1.10)=\$200(1.10)=\$220$. The expected death-benefit amount is $0.20(\$500)=\$100$, and $p_x=0.80$. Hence

{}_1V_x=\frac{\$220-\$100}{0.80}=\$150.

Second year. The amount accumulated to time 2 is $(\$150+\$115-\$15)(1.10)=\$250(1.10)=\$275$. Here $p_{x+1}=0.90$ and the expected death-benefit amount is $0.10(\$500)=\$50$. Therefore,

{}_2V_x=\frac{\$275-\$50}{0.90}=\$250.

Backward check. From time 2, the expected time-2 requirement is $0.10(\$500)+0.90(\$250)=\$275$. Discounting and applying the year-2 cash flow gives ${}_1V_x=\$275/1.10-\$115+\$15=\$250-\$100=\$150$. For year 1, the expected time-1 requirement is $0.20(\$500)+0.80(\$150)=\$220$. Thus ${}_0V_x=\$220/1.10-\$140+\$20=\$200-\$120=\$80$. Both backward calculations recover the given and calculated reserves. The common mistake avoided is treating an expense as a premium: the expense must be subtracted from the amount available, not added to it.

Checks, Limits, and Common Traps

Forward and backward calculations are especially useful as mutual checks. After calculating a sequence in one direction, substitute each adjacent pair of reserves into the original annual balance equation. A correct pair must satisfy that equation using the same mortality, interest, benefit, premium, expense, and timing assumptions.

Mixing reserve timing: The equation here uses a reserve immediately before the premium. If a given reserve is immediately after the premium, first translate the cash-flow balance to that convention; do not insert it unchanged.
Using the wrong probability: The death benefit is weighted by $q_{x+k}$; the surviving reserve is weighted by $p_{x+k}=1-q_{x+k}$. These probabilities apply to the policy in force at the beginning of that year.
Misplacing interest: Premiums and expenses at time $k$ are included before the factor $1+i$. A benefit paid at the end of the year is not accumulated for an additional year.
Dividing by survival in a backward calculation: The backward formula does not divide by $p_{x+k}$. Division by survival is needed only to isolate a later reserve in the forward formula.
Ignoring model scope: This equation assumes annual cash flows at the specified times and a single annual death benefit. It must be adjusted if benefits, premiums, or expenses occur at different times or if the stated reserve uses another convention.

Key takeaway: Write the annual balance with its timing labels before calculating. Forward recursion accumulates the earlier reserve and time-$k$ net cash flow, then isolates the survivor reserve. Backward recursion discounts the expected later requirement, then removes the premium and adds back the expense.

Check Your Understanding

For each problem, use the before-premium reserve convention defined above. Amounts are hypothetical.

A policy has ${}_0V_x=\$40$, $P_0=\$60$, no expense, $i=5\%$, $q_x=0.20$, and a year-end death benefit of $\$250$. Calculate ${}_1V_x$. (Answer: $\$68.75$)
A policy has $i=0$, $P_0=P_1=\$50$, no expenses, a $\$400$ year-end death benefit in each year, $q_x=0.25$, $q_{x+1}=0.20$, and ${}_2V_x=\$160$. Calculate ${}_1V_x$ and then ${}_0V_x$. (Answer: ${}_1V_x=\$158$, ${}_0V_x=\$168.50$)
For one year, $i=4\%$, $q_x=0.10$, $B_1=\$600$, $P_0=\$80$, $e_0=\$10$, and ${}_1V_x=\$200$. Calculate ${}_0V_x$, rounded to the nearest cent. (Answer: $\$160.77$)
A two-year policy has $i=10\%$, a $\$1{,}000$ year-end death benefit, $P_0=P_1=\$100$, and no expenses. Let $q_x=0.20$, $q_{x+1}=0.10$, and ${}_0V_x=\$100$. Calculate ${}_1V_x$ and ${}_2V_x$. (Answer: ${}_1V_x=\$25$, ${}_2V_x=\$41.6667$)
A policy has $i=5\%$, a $\$500$ year-end death benefit, $q_x=0.20$, $q_{x+1}=0.10$, and ${}_2V_x=\$250$. In year 1, $P_1=\$120$ and $e_1=\$20$; at time 0, $P_0=\$130$ and $e_0=\$10$. Calculate ${}_1V_x$ and ${}_0V_x$, rounding to cents. (Answer: ${}_1V_x=\$161.90$, ${}_0V_x=\$98.59$)

What You'll Learn