Return-of-Premium Benefits in Reserve Recursions

Why a Premium Refund Changes the Recursion

A return-of-premium provision makes the death benefit depend on the premiums already paid. The insurer therefore cannot treat the death benefit as a fixed amount across policy years: the refund component generally grows with duration. The key is to track that component separately from the reserve and to identify exactly which premiums the contract refunds.

This tutorial uses a discrete annual model. Premiums are due at the beginning of each policy year while the insured is alive, and a death benefit is paid at the end of the year of death. The reserve at time $k$, denoted $V_k$, is measured immediately before the premium due at time $k$. Thus $V_k+P_k$ is available just after that premium is received. The end-of-year reserve $V_{k+1}$ is measured immediately before the next premium.

Definition: Let $P_k$ be the premium due at time $k$, and let $R_k$ be the total premiums paid before that time. With premiums paid at times $0,1,\ldots$ while the policy is in force, $$ R_0=0,\qquad R_{k+1}=R_k+P_k. $$ If the death benefit in policy year $k+1$ is a base amount $B_{k+1}$ plus a refund of premiums paid to date, then the benefit is $b_{k+1}=B_{k+1}+R_{k+1}$. The premium due at time $k$ is included because it has been paid before a death benefit at the end of that year is determined.

The amount $R_{k+1}$ is the sum of premiums actually paid, not the accumulated value of those premiums with interest. A contract could specify an interest-adjusted refund or define the refund differently, but that would be a different benefit specification. The recursion below applies to a refund of premiums without interest.

The Discrete Reserve Recursion

Assume the insured is age $x+k$ at time $k$, the annual effective interest rate is $i$, and death during the year has probability $q_{x+k}$. Write ${}_1p_{x+k}=1-q_{x+k}$ for the probability of surviving the year. Conditional on being alive just before the premium at time $k$, the insurer receives $P_k$. At the end of the year it pays $b_{k+1}$ if death occurs, or holds the reserve $V_{k+1}$ if the insured survives.

Theorem: Under the annual timing and reserve convention just stated, the recursion is $$ (V_k+P_k)(1+i) =q_{x+k}\bigl(B_{k+1}+R_{k+1}\bigr) +{}_1p_{x+k}V_{k+1}. $$ Equivalently, with $v=1/(1+i)$, $$ V_k =v\left[q_{x+k}\bigl(B_{k+1}+R_{k+1}\bigr) +{}_1p_{x+k}V_{k+1}\right]-P_k. $$

The first form describes the one-year balance at time $k+1$: the premium and reserve at time $k$ earn interest for one year, and the resulting amount must cover either the death claim or the survivor reserve. The second form isolates the reserve before the premium. If there is no refund provision, replace $B_{k+1}+R_{k+1}$ by the specified death benefit. If the premium timing or the date at which the reserve is measured differs, the premium and reserve terms must be adjusted.

Why the Refund Accumulates This Way

The cumulative refund is a deterministic policy-duration amount, conditional on the policy remaining in force and the scheduled premiums being paid. It is not a second reserve. The recursion follows directly from the one-year outcomes.

Record the premium at time $k$.
The insurer has $V_k+P_k$ immediately after the beginning-of-year premium is paid.

Accumulate for one year.
At the effective annual rate $i$, the amount becomes $(V_k+P_k)(1+i)$.

Separate death and survival outcomes.
Death has probability $q_{x+k}$ and requires payment $B_{k+1}+R_{k+1}$; survival has probability ${}_1p_{x+k}$ and requires reserve $V_{k+1}$.

Taking the expected amount required at the end of the year gives $$ q_{x+k}(B_{k+1}+R_{k+1})+{}_1p_{x+k}V_{k+1}. $$ Equating this with the accumulated amount proves the recursion. This balance argument assumes the stated annual model: the death claim is paid at year-end, there are no expenses in the equation, and the premium is received at the beginning of the year.

Worked Example: A Two-Year Term With Level Premiums

A hypothetical two-year term policy has annual effective interest $i=5\%$, annual mortality probabilities $q_x=0.10$ and $q_{x+1}=0.20$, and a base death benefit of $1,000 in either year. A level premium of $100 is due at times 0 and 1 while the insured is alive. The end-of-year death benefit refunds all premiums paid, with no interest. Find $V_1$ and $V_0$, where reserves are measured before each scheduled premium. The terminal reserve is $V_2=0$.

Method. First accumulate the refund by policy year, then apply the recursion backwards from the terminal reserve. The common mistake to avoid is refunding only the premiums paid before the current year: the premium due at the beginning of the year is also paid before a year-end claim.

In year 1, one premium has been paid, so $R_1=100$ and the death benefit is $1{,}000+100=1{,}100$. In year 2, two premiums have been paid, so $R_2=200$ and the benefit is $1{,}000+200=1{,}200$. At time 1, $$ V_1 =\frac{0.20(1{,}200)+0.80(0)}{1.05}-100 =\frac{240}{1.05}-100 =228.5714-100 =128.5714. $$ Thus $V_1=\$128.57$, rounded to the nearest cent.

At time 0, the probabilities are $q_x=0.10$ and ${}_1p_x=0.90$, and the year-1 benefit is $1,100. Therefore, $$ \begin{aligned} V_0 &=\frac{0.10(1{,}100)+0.90(128.5714)}{1.05}-100\\ &=\frac{110+115.7143}{1.05}-100\\ &=214.9660-100\\ &=114.9660. \end{aligned} $$ Hence $V_0=\$114.97$, rounded to the nearest cent.

Check. The expected present value of claims is $110/1.05+216/1.05^2=\$300.68$. The expected present value of premiums is $100+0.90(100)/1.05=\$185.71$. Their difference is $\$114.97$, agreeing with $V_0$.

Duration-Varying Premiums

The same method applies when premiums vary by duration. The refund is then the sum of the premiums paid so far, not the current premium multiplied by the number of years. This distinction matters both in the death benefit and in the reserve recursion.

Worked Example: Refund of Unequal Premiums

A hypothetical two-year policy has $i=5\%$, $q_x=0.10$, $q_{x+1}=0.20$, and a $5,000 base death benefit. The premiums are $P_0=\$200$ and $P_1=\$300$. Each year’s death benefit refunds all premiums paid to date, without interest. Find the reserves before the premiums at times 1 and 0. Assume $V_2=0$.

Method. Build the cumulative premiums first. In year 1, the refund is $200, giving a benefit of $5,200. In year 2, it is $200+$300=$500, giving a benefit of $5,500. Do not use $2P_1$ for the second-year refund: the two premiums are unequal.

Starting at time 1, $$ \begin{aligned} V_1 &=\frac{0.20(5{,}500)+0.80(0)}{1.05}-300\\ &=\frac{1{,}100}{1.05}-300\\ &=1{,}047.6190-300\\ &=747.6190. \end{aligned} $$ So $V_1=\$747.62$, rounded to the nearest cent.

At time 0, the year-1 benefit is $5,200: $$ \begin{aligned} V_0 &=\frac{0.10(5{,}200)+0.90(747.6190)}{1.05}-200\\ &=\frac{520+672.8571}{1.05}-200\\ &=1{,}136.0544-200\\ &=936.0544. \end{aligned} $$ Thus $V_0=\$936.05$, rounded to the nearest cent.

Check. The present value of expected claims is $520/1.05+990/1.05^2=\$1{,}393.20$. The present value of premiums is $200+0.90(300)/1.05=\$457.14$. The difference is $\$936.05$, as obtained recursively.

When the Premium Is Unknown

Under a net premium equivalence calculation, the premium can affect both sides of the equation: it funds the policy, but it is also part of the refund if death occurs. The premium must therefore remain in the benefit expression until the equation is solved. In a one-year contract with no terminal benefit, a single premium $P$ is paid at time 0 and the death benefit is $B+P$. If the reserve at issue is zero, the recursion gives $P(1+i)=q_x(B+P)$.

Worked Example: Solving a One-Year Net Premium

A hypothetical one-year policy pays $9,600 plus a refund of the single premium if death occurs during the year. The annual effective interest rate is $i=4\%$, and $q_x=0.20$. The premium $P$ is paid at time 0, and the issue reserve is zero. Find the net premium.

Method. Use the one-year recursion with terminal reserve zero. The refund contains $P$, so omitting it from the benefit would understate expected claims.

The recursion is $$ (0+P)(1.04)=0.20(9{,}600+P)+0.80(0). $$ Expanding and collecting terms gives $$ 1.04P=1{,}920+0.20P,\qquad 0.84P=1{,}920,\qquad P=\frac{1{,}920}{0.84}=2{,}285.7143. $$ The net premium is $2,285.71, rounded to the nearest cent.

Check. The accumulated premium is $1.04(2{,}285.7143)=\$2{,}377.14$. The expected year-end claim is $0.20(9{,}600+2{,}285.7143)=\$2{,}377.14$. The two sides agree.

Common Timing and Refund Errors

A quick way to catch mistakes is to write the premium dates and the claim date before writing the recursion. For a death at the end of year $k+1$, the premium at time $k$ has already been paid under the timing used here. If a contract instead refunds only premiums paid before the current year’s premium, the benefit in year $k+1$ would use $R_k$, not $R_{k+1}$; that contractual definition must be stated and applied consistently.

Leaving out the current year’s premium. Under beginning-of-year premium timing and an end-of-year death claim, the refund includes $P_k$. The year-$k+1$ benefit uses $R_{k+1}=R_k+P_k$.
Adding interest to the refund without a contract provision. A refund of premiums paid is not automatically an accumulated account. Use only the benefit specified.
Refunding the wrong amount when premiums vary. The cumulative refund is $P_0+\cdots+P_k$, not $(k+1)P_k$ unless the premiums are level.
Using inconsistent reserve timing. Here $V_k$ is before the premium, so the insurer starts the year with $V_k+P_k$. Omitting $P_k$, or adding it twice, changes the recursion.

For example, in the first worked example, treating the first-year death benefit as $1,000 rather than $1,100 would omit the premium paid at time 0. The incorrect initial recursion would be $$ V_0^{\text{wrong}} =\frac{0.10(1{,}000)+0.90(128.5714)}{1.05}-100 =\$105.44. $$ The correct benefit includes the $100 refund, giving $V_0=\$114.97$. The $9.52 difference is the present value of the omitted expected first-year refund, $0.10(100)/1.05$.

Key takeaway: For a discrete policy with a refund of premiums paid, determine the cumulative refund at each possible claim duration before calculating reserves. With premiums due at the beginning of the year and claims paid at year-end, the recursion uses the current premium in both the funds available, $V_k+P_k$, and the refund amount, $R_{k+1}$.

Check Your Understanding

Use beginning-of-year premiums, end-of-year death benefits, and reserves measured before the premium unless a problem states otherwise. Each refund is of premiums paid, without interest.

A one-year term policy has $i=0$, $q_x=0.25$, base benefit $6,000, and premium $500 at time 0. The terminal reserve is zero. Find $V_0$. (Answer: $1,125.00)
A two-year policy has $i=0$, $q_x=0.10$, $q_{x+1}=0.20$, base benefit $4,000, and premiums $P_0=\$100$, $P_1=\$200$. Find $V_1$ and $V_0$, given $V_2=0$. (Answer: $V_1=\$660.00$, $V_0=\$904.00$)
A one-year policy has base benefit $9,000, $q_x=0.10$, $i=5\%$, and a single net premium $P$ refunded upon death. Find $P$ using a zero issue reserve. (Answer: $947.37)
A two-year policy has $i=0$, $q_x=0.10$, $q_{x+1}=0.20$, and level net premium $P$ at times 0 and 1 while in force. The base death benefit is $5,000, and all premiums paid are refunded. Find $P$ using equivalence. (Answer: $972.22)
A two-year policy has $i=4\%$, $q_x=0.05$, $q_{x+1}=0.10$, base benefit $10,000, and premiums $P_0=\$300$, $P_1=\$500$. Find $V_1$ and $V_0$, given $V_2=0$. (Answer: $V_1=\$538.46$, $V_0=\$687.06$)

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