Recursions for Beginning- and End-of-Year Benefits

Why Benefit Timing Changes the Recursion

A one-year reserve recursion balances funds available at the start of a policy year against payments and reserves at the end of that year. The mortality probabilities and interest rate may be unchanged, yet the recursion differs depending on when the death benefit is paid. A benefit paid at the beginning of the year of death is not discounted for that year; a benefit paid at the end is.

The distinction is easiest to see when reserves are measured consistently. In this tutorial, $V_k$ is the reserve immediately before the premium due at time $k$, and $V_{k+1}$ is the reserve immediately before the next annual premium. A premium $P$ is collected at time $k$ from each policyholder alive then. The benefit $B$ is payable upon death during the year. We assume a deterministic effective annual interest rate $i$, no expenses or lapses, and annual mortality probabilities that apply over the year.

Definition: For a life aged $x$ at the start of the year, $q_x$ is the probability of death during that year and $p_x=1-q_x$ is the probability of survival. The discount factor is $v=1/(1+i)$. A beginning-of-year death benefit is paid at the start of the year in which death occurs; an end-of-year death benefit is paid at the end of that year.

The beginning-of-year convention is a discrete timing assumption: deaths occurring during the year are represented as claims paid at the year’s start. It is not a claim about the exact time of death. The premium is due at the start of the year, and the recursion below assumes it has been collected before the beginning-of-year claim is paid.

The Two One-Year Reserve Equations

The next equations are one-year expected-value balances. The reserve at time $k$, together with the premium collected then, must equal the present value of the expected claim and the reserve required for survivors at time $k+1$. Mortality probabilities weight the death and survival outcomes; discounting reflects the time at which each payment or reserve is needed.

Theorem: Under the assumptions above, for a death benefit $B$ and premium $P$, the one-year recursions are:

Benefit paid at the beginning of the year of death: $V_k+P=q_xB+p_xvV_{k+1}$.

Benefit paid at the end of the year of death: $V_k+P=v(q_xB+p_xV_{k+1})$.

In both equations, the survivor reserve is needed at time $k+1$, so its value is discounted by $v$. The difference is the death claim: in the beginning-of-year equation it is paid at time $k$, while in the end-of-year equation it is paid at time $k+1$ and therefore discounted.

Start with available funds.
At time $k$, the reserve before premium plus the premium collected is $V_k+P$.

Separate death and survival.
Death has probability $q_x$; survival has probability $p_x$. A death claim is $B$, while a survivor requires reserve $V_{k+1}$ at time $k+1$.

Discount each amount to time $k$.
Discount a payment made at time $k+1$, but do not discount a payment made at time $k$. Weight each outcome by its probability.

Why the Equations Hold

The timing rule follows directly from valuing each possible outcome at the same date. For a beginning-of-year benefit, death during the year leads to a claim $B$ at time $k$, while survival leads to the reserve $V_{k+1}$ one year later. Their probability-weighted values at time $k$ are $q_xB$ and $p_xvV_{k+1}$, respectively. Equating their sum to funds available gives the first recursion.

V_k+P=q_xB+p_xvV_{k+1}.

For an end-of-year benefit, both outcomes require funds at time $k+1$: the death claim $B$ if death occurs and the reserve $V_{k+1}$ if the insured survives. Their expected amount at time $k+1$ is $q_xB+p_xV_{k+1}$. Discounting that amount one year gives the second equation:

V_k+P=v(q_xB+p_xV_{k+1}).

This derivation also shows precisely when the equations do not apply as written. If benefits are paid at a different time, premiums are due at a different time, or expenses and other cash flows are included, the corresponding amounts must be valued at their actual payment dates. The reserve timing convention must also remain consistent from one duration to the next.

Worked Applications

Worked Example: Comparing Benefit Timing

A hypothetical policy has a death benefit of $10,000 and an annual premium of $500. At the start of a particular year, $q_x=0.10$, $p_x=0.90$, and $i=0.05$. The reserve immediately before the next premium is $2,000. Calculate the reserve immediately before the current premium under each benefit timing.

Method. Use the backward recursion for each payment convention. The parameters are $B=10{,}000$, $P=500$, $V_{k+1}=2{,}000$, $q_x=0.10$, $p_x=0.90$, and $v=1/1.05=0.95238095$.

Beginning-of-year benefit. The death claim is paid at the start of the year, so it is not discounted:

\begin{aligned} V_k+500 &=0.10(10{,}000)+0.90(0.95238095)(2{,}000)\\ &=1{,}000+1{,}714.28571\\ &=2{,}714.28571,\\ V_k&=2{,}714.28571-500=2{,}214.28571. \end{aligned}

End-of-year benefit. The death claim and survivor reserve are both valued one year later, so both are discounted:

\begin{aligned} V_k+500 &=0.95238095\left[0.10(10{,}000)+0.90(2{,}000)\right]\\ &=0.95238095(1{,}000+1{,}800)\\ &=2{,}666.66666,\\ V_k&=2{,}666.66666-500=2{,}166.66666. \end{aligned}

Rounded to the nearest cent, the reserves are $2,214.29 for the beginning-of-year benefit and $2,166.67 for the end-of-year benefit. The beginning-of-year reserve is larger because the death claim is not discounted. Substitution into each original equation reproduces the calculated total funds, checking both results. The method avoids applying the end-of-year discount to a claim paid at the beginning of the year.

Worked Example: Calculating Reserves Backward Over Two Years

A hypothetical two-year contract pays $5,000 upon death, with the claim treated as payable at the beginning of the year of death. The premium is $300 at each year’s start. In both years, $q_x=0.20$, $p_x=0.80$, and $i=0.25$. The reserve immediately before the premium at time 2 is zero. Find the reserves immediately before the premiums at times 1 and 0.

Method. Work backward one policy year at a time using $v=1/1.25=0.80$ and the beginning-of-year recursion $V_k=q_xB+p_xvV_{k+1}-P$.

At time 1, $V_2=0$, so:

\begin{aligned} V_1+300&=0.20(5{,}000)+0.80(0.80)(0),\\ V_1+300&=1{,}000,\\ V_1&=700. \end{aligned}

At time 0, use the reserve just found as the next-year reserve:

\begin{aligned} V_0+300&=0.20(5{,}000)+0.80(0.80)(700),\\ V_0+300&=1{,}000+448=1{,}448,\\ V_0&=1{,}148. \end{aligned}

The reserves are $V_1=$700$ and $V_0=$1,148$. Checking forward, the time-0 balance is $1{,}148+300=1{,}448=1{,}000+0.64(700)$; the time-1 balance is $700+300=1{,}000$, equal to the expected beginning-of-year claim because $V_2=0$. This backward method avoids using the reserve at time 2 in place of the reserve at time 1.

Worked Example: Solving Forward for the Next Reserve

A hypothetical policy has a beginning-of-year death benefit of $5,000. At the start of the year, the reserve before premium is $1,000 and the premium is $300. Assume $q_x=0.20$, $p_x=0.80$, and $i=0.25$. Calculate the next reserve, then calculate the next reserve if the same benefit were instead paid at year-end.

Method. Rearrange each recursion to solve for $V_{k+1}$. Here $v=1/1.25=0.80$, and $V_k+P=1{,}000+300=1{,}300$.

Beginning-of-year benefit:

\begin{aligned} 1{,}300&=0.20(5{,}000)+0.80(0.80)V_{k+1},\\ 1{,}300-1{,}000&=0.64V_{k+1},\\ V_{k+1}&=\frac{300}{0.64}=468.75. \end{aligned}

End-of-year benefit:

\begin{aligned} 1{,}300&=0.80\left[0.20(5{,}000)+0.80V_{k+1}\right],\\ \frac{1{,}300}{0.80}&=1{,}000+0.80V_{k+1},\\ 1{,}625-1{,}000&=0.80V_{k+1},\\ V_{k+1}&=\frac{625}{0.80}=781.25. \end{aligned}

The required next reserves are $468.75 under beginning-of-year timing and $781.25 under end-of-year timing. Substituting either value into its respective equation gives $V_k+P=$1,300$, which checks the calculations. The method avoids rearranging the end-of-year equation as though the death benefit were paid at the start of the year.

Common Timing Error and a Useful Check

A frequent error is to discount every death benefit automatically. For example, in the beginning-of-year case of the first worked example, an incorrect calculation would use $v[q_xB+p_xV_{k+1}]$, giving $2,166.67 for the reserve. That is the end-of-year recursion, not the beginning-of-year recursion. The correct calculation is $q_xB+p_xvV_{k+1}-P$, giving $2,214.29. The correction is to identify the payment date of each cash flow before applying the discount factor.

For the same $B$, $P$, $q_x$, $p_x$, and $V_{k+1}$, the two reserves differ only in the timing of the death claim. Subtracting the end-of-year equation from the beginning-of-year equation gives:

V_k^{\text{begin}}-V_k^{\text{end}} =q_xB-vq_xB =q_xB(1-v).

The survivor-reserve terms and premium cancel. When $i>0$, $v<1$, so a positive death benefit produces a higher reserve under beginning-of-year payment. This is a useful diagnostic, not a substitute for checking the cash-flow timing: if the computed beginning-of-year reserve is lower under these same assumptions, revisit the discounting and signs.

These recursions are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. In applications, first state when the reserve is measured, then locate each premium, claim, and next reserve on the timeline. That discipline prevents the common mistake of using the right probabilities with the wrong discount factor.

Check Your Understanding

For each problem, use the stated reserve convention: $V_k$ and $V_{k+1}$ are measured immediately before the premiums at their respective times.

A hypothetical contract has a beginning-of-year death benefit of $6,000, premium $200, $q_x=0.12$, $p_x=0.88$, $i=0.04$, and $V_{k+1}=$1,500$. Find $V_k$, rounded to the nearest cent. (Answer: $1,789.23)
A hypothetical policy pays a $3,000 death benefit at year-end. Given $q_x=0.20$, $p_x=0.80$, $i=0.25$, premium $100, and $V_{k+1}=$500$, calculate $V_k$. (Answer: $700.00)
A hypothetical policy pays a $4,000 death benefit at year-end. Given $V_k=$900$, premium $100, $q_x=0.10$, $p_x=0.90$, and $i=0.05$, solve for $V_{k+1}$, rounded to the nearest cent. (Answer: $722.22)
For a hypothetical beginning-of-year benefit, $B=$5,000$, $q_x=0.30$, $p_x=0.70$, $i=0.10$, premium $250, and $V_{k+1}=$1,000$. Calculate $V_k$, then find the reserve if the same benefit were paid at year-end. (Answer: beginning-of-year reserve $1,886.36; end-of-year reserve $1,750.00)

What You'll Learn