Finding a Level Premium from a Prescribed Reserve

Using Adjacent Reserves to Find a Premium

A reserve recursion is usually used to find a reserve when the premium is known. The same equation can be rearranged to answer the reverse question: what level premium is consistent with specified reserves, assumptions, and expenses? This is useful when an actuarial calculation supplies reserve values at two successive policy durations and asks for the premium that makes the one-year balance hold.

The calculation depends on timing. In this tutorial, $V_k$ is the reserve immediately before the scheduled premium at time $k$, and $V_{k+1}$ is the reserve immediately before the next scheduled premium. The premium and the stated expenses are paid or incurred at time $k$. A death benefit is paid at the end of the year of death; if the insured survives the year, the next reserve is required at time $k+1$. These conventions are part of the model, not details to infer after calculating.

Definition: Let $P$ be the level annual premium, $i$ the effective annual interest rate, ${}_1q_{x+k}$ the probability of death during the year, and ${}_1p_{x+k}=1-{}_1q_{x+k}$ the probability of survival. Let $B$ be the end-of-year death benefit, and let $e$ be an expense paid at time $k$. Under the timing just stated, the one-year reserve recursion is $$ (V_k+P-e)(1+i)={}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}. $$

The left side is the amount available at the start of the year, accumulated with interest. The right side is the expected amount required at year-end: the death benefit weighted by the probability of death, plus the next reserve weighted by the probability of survival. The recursion assumes the stated benefits and reserves apply to the respective outcomes and that the one-year interest and mortality assumptions are appropriate for this policy year.

Rearranging the Recursion

When $e$ is a known fixed expense, rearranging gives the premium directly. First calculate the expected year-end requirement, then discount it one year, and compare it with the existing reserve and expense.

P=e+\frac{{}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}}{1+i}-V_k.

Expenses may instead include a fixed amount and a percentage of premium. Write the expense as $e=a+\beta P$, where $a$ is the fixed expense per policy and $\beta$ is the proportion of premium consumed by expense. Substitution into the recursion gives

(V_k+P-a-\beta P)(1+i) ={}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}.

Collect the premium terms and divide by $1+i$:

(1-\beta)P =\frac{{}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}}{1+i}-V_k+a, \qquad P= \frac{\dfrac{{}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}}{1+i}-V_k+a}{1-\beta}.

This solution requires $\beta\ne 1$. In the usual premium-setting context, $0\leq\beta<1$, so each dollar of premium contributes a positive amount after proportional expense. A negative calculated premium is an algebraic result, but it normally signals that the specified reserves and assumptions do not support a nonnegative premium under the stated model.

Why the Expected Requirement Appears

The recursion follows by conditioning on whether death occurs during the year. If death occurs, probability ${}_1q_{x+k}$, the insurer needs $B$ at year-end. If survival occurs, probability ${}_1p_{x+k}$, the insurer needs $V_{k+1}$. Thus the expected year-end amount is ${}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}$. The assets available immediately after the premium and expense are $V_k+P-e$; accumulating those assets for one year yields the left side. Equating the accumulated assets to the expected requirement gives the recursion. This is a one-year balance, so it does not require projecting every future cash flow separately once the adjacent reserves are supplied.

Identify timing and inputs.
Confirm that the adjacent reserves are measured at the same specified points in the premium cycle, and identify the year’s death benefit and expense timing.

Find the year-end requirement.
Calculate ${}_1q_{x+k}B+{}_1p_{x+k}V_{k+1}$, using probabilities that sum to one.

Discount and solve.
Divide the requirement by $1+i$, then use the appropriate fixed-expense or premium-proportional formula.

Check the balance.
Substitute the unrounded premium into the start-of-year asset amount and verify that accumulation at interest matches the expected year-end requirement.

Worked Examples

Worked Example: Fixed Renewal Expense

A hypothetical one-year policy has a reserve of $\$500$ immediately before the premium at time $k$. The reserve immediately before the next premium is zero. The death benefit is $\$10{,}000$, the probability of death during the year is $0.10$, and the effective annual interest rate is $10\%$. A fixed renewal expense of $\$20$ is incurred at time $k$. Find the level premium for this year.

Method. Use the expected year-end requirement and the fixed-expense rearrangement. Here ${}_1p_{x+k}=1-0.10=0.90$.

The death-outcome contribution is $0.10(\$10{,}000)=\$1{,}000$. The survival-outcome contribution is $0.90(\$0)=\$0$. Therefore, the expected year-end requirement is $\$1{,}000+\$0=\$1{,}000$.

Discount that amount one year: $\$1{,}000/1.10=\$909.0909\ldots$. The premium is

P=20+\frac{1{,}000}{1.10}-500 =20+909.0909\ldots-500 =\$429.0909\ldots.

Rounded to the nearest cent, the premium is $\$429.09$. To check, using the unrounded premium, the amount after the premium and fixed expense is $\$500+\$429.0909\ldots-\$20=\$909.0909\ldots$. Accumulating gives $\$909.0909\ldots(1.10)=\$1{,}000$, exactly the expected requirement. This check avoids treating the fixed expense as an expense at year-end.

Worked Example: Fixed and Premium-Proportional Expenses

A hypothetical policy has $V_k=\$3{,}000$ and $V_{k+1}=\$3{,}300$. The annual effective interest rate is $2\%$, the probability of death is $0.01$, and the end-of-year death benefit is $\$60{,}000$. At time $k$, the insurer incurs a fixed expense of $\$25$ plus an expense equal to $4\%$ of the premium. Find the level premium.

Method. First calculate the probability-weighted year-end requirement. Survival probability is $1-0.01=0.99$. The death contribution is $0.01(\$60{,}000)=\$600$, and the survival contribution is $0.99(\$3{,}300)=\$3{,}267$. Thus the requirement is $\$600+\$3{,}267=\$3{,}867$.

The amount required before interest is $\$3{,}867/1.02=\$3{,}791.176470\ldots$. The expense is $25+0.04P$, so the pre-interest assets are $3{,}000+P-25-0.04P=2{,}975+0.96P$. Equate the two amounts:

2{,}975+0.96P=3{,}791.176470\ldots, \qquad 0.96P=816.176470\ldots, \qquad P=\$850.183823\ldots.

Rounded to the nearest cent, the premium is $\$850.18$. For the check, use the unrounded premium. The amount available before interest is $3{,}000+850.183823\ldots-25-0.04(850.183823\ldots)=\$3{,}791.176470\ldots$, or $\$3{,}791.18$ to the nearest cent. It accumulates to $\$3{,}791.176470\ldots(1.02)=\$3{,}867$, the expected year-end requirement.

Common mistake avoided: omitting the $4\%$ premium expense would give $P=3{,}791.176470\ldots-3{,}000+25=\$816.18$, which is too low. The premium must be reduced by its own proportional expense in the recursion.

Worked Example: Solving with a Different Expense Proportion

A hypothetical policy has $V_k=\$3{,}500$, $V_{k+1}=\$4{,}000$, an annual effective interest rate of $4\%$, a $0.02$ probability of death, and a $\$50{,}000$ death benefit at year-end. Expenses at time $k$ are $\$30$ plus $5\%$ of the premium. Determine the premium and check the result.

Method. Use the general formula with $a=30$ and $\beta=0.05$. Survival probability is $0.98$. The expected year-end requirement is $0.02(\$50{,}000)+0.98(\$4{,}000)=\$1{,}000+\$3{,}920=\$4{,}920$.

Discounting gives $\$4{,}920/1.04=\$4{,}730.769230\ldots$. Therefore,

P= \frac{4{,}730.769230\ldots-3{,}500+30}{1-0.05} = \frac{1{,}260.769230\ldots}{0.95} =\$1{,}327.125506\ldots.

Rounded to the nearest cent, the premium is $\$1{,}327.13$. Using the unrounded value, the expense is $\$30+0.05(\$1{,}327.125506\ldots)=\$96.356275\ldots$. The amount available before interest is $\$3{,}500+\$1{,}327.125506\ldots-\$96.356275\ldots=\$4{,}730.769230\ldots$. After interest it is $\$4{,}730.769230\ldots(1.04)=\$4{,}920$, matching the expected requirement.

Common Traps and Interpretation

A premium obtained from adjacent reserves is conditional on the assumptions used in the recursion. If the benefit is paid at a different time, the one-year accumulation or discounting must reflect that timing. If the next reserve is defined immediately after rather than immediately before the next premium, the survivor cash flow at time $k+1$ must also be handled consistently. Do not insert an extra premium into the formula without first checking how the reserve is defined.

Using the wrong probability on a cash flow. The death benefit is weighted by the probability of death; the next reserve is weighted by the probability of survival. Reversing these weights changes the expected requirement.
Forgetting to discount the year-end requirement. The beginning-of-year assets are compared with the year-end requirement only after accumulation. Equivalently, discount the requirement by $1+i$.
Omitting proportional expenses. If expense includes $\beta P$, the coefficient on $P$ is $1-\beta$, not $1$.
Rounding too early. Carry the unrounded premium through the asset check. Rounding the final premium to cents can create a small difference in the check, but it should not change the stated reserve assumptions.

The result is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The central skill is not memorizing one rearrangement; it is identifying the time points, constructing the two outcome-weighted year-end amounts, and incorporating every expense exactly once.

Check Your Understanding

For each problem, use the stated timing convention: reserves are immediately before premiums, expenses occur at the start of the year, and death benefits are paid at year-end.

A policy has $V_k=\$900$, $V_{k+1}=\$1{,}200$, death probability $0.03$, death benefit $\$20{,}000$, interest $5\%$, and a fixed expense of $\$15$. Find $P$. (Answer: $\$795.00$)
A policy has $V_k=\$2{,}500$, $V_{k+1}=\$3{,}000$, death probability $0.01$, death benefit $\$40{,}000$, and interest $2\%$. Expenses are $\$20$ plus $4\%$ of premium. Find $P$, to the nearest cent. (Answer: $\$858.25$)
A policy has $V_k=\$1{,}000$, $V_{k+1}=\$2{,}000$, death probability $0.05$, death benefit $\$20{,}000$, and interest $5\%$. Expenses are $\$40$ plus $10\%$ of premium. Find $P$, to the nearest cent. (Answer: $\$2{,}002.12$)
A policy has $V_k=\$2{,}200$, $V_{k+1}=\$2{,}500$, death probability $0.04$, death benefit $\$15{,}000$, and interest $4\%$. Expenses are $\$50$ plus $2\%$ of premium. Find $P$, to the nearest cent. (Answer: $\$749.61$)
A policy has $V_k=\$5{,}000$, $V_{k+1}=\$4{,}000$, death probability $0.01$, death benefit $\$10{,}000$, and interest $2\%$. Expenses are $\$20$ plus $5\%$ of premium. Find the algebraic solution and state what its sign indicates. (Answer: $-\$1{,}052.22$; the specified assumptions and reserves do not support a nonnegative premium in this model.)

What You'll Learn