Using Life-Table Counts for Retrospective Reserves

Why Life-Table Counts Are Useful

A retrospective reserve measures the accumulated past premiums available, after past claims, for policies still in force at a valuation date. When a life table gives counts rather than probabilities, it can be used directly to calculate those cohort totals. The key is to distinguish who pays or receives a cash flow from who remains in force at the valuation date.

A count of policies alive at a given age weights premiums due at that age. A count of deaths during a policy year weights death benefits paid at the end of that year. After accumulating the resulting cash flows to the valuation date, divide by the number of policies surviving to that date. That last step converts a cohort total into a reserve per survivor.

Definition: Let $\ell_{x+j}$ be the number alive at exact age $x+j$ in a life-table cohort, and let $d_{x+j}=\ell_{x+j}-\ell_{x+j+1}$ be the number dying between ages $x+j$ and $x+j+1$. A retrospective reserve at duration $k$, per policy surviving to age $x+k$, is the accumulated value at time $k$ of premiums received before time $k$ minus claims paid before time $k$, divided by $\ell_{x+k}$.

Assumptions and the Life-Table Formula

Use the following annual, discrete model. A cohort of $\ell_x$ policies is issued at time 0. A premium $P_j$ is due at the beginning of policy year $j+1$, at time $j$, if the policy is then in force. A death benefit $B_{j+1}$ is paid at the end of that year, at time $j+1$, if death occurs during the year. Interest is earned at the effective annual rate $i$. The reserve at time $k$ is measured after the cash flows through time $k$, but before any premium due at time $k$.

The formula assumes that the life-table counts describe the cohort under consideration, that benefits and premiums follow the stated timing, and that there are no other cash flows. If the contract has expenses, maturity payments, or different payment timing, those cash flows must be included at their actual times. The formula below does not automatically account for them.

{}_kV_x = \frac{ \displaystyle\sum_{j=0}^{k-1} P_j\ell_{x+j}(1+i)^{k-j} - \displaystyle\sum_{j=0}^{k-1} B_{j+1}d_{x+j}(1+i)^{k-j-1} }{ \ell_{x+k} }

If the annual premium is level, write $P_j=P$ for all applicable years. A premium paid at time $j$ earns interest for $k-j$ years before time $k$, which explains its factor $(1+i)^{k-j}$. A claim paid at time $j+1$ earns interest for only $k-j-1$ years. These exponents differ by one because premiums are due at the beginning of the year and death benefits at its end.

Why the Count Formula Works

Each life-table count contributes the cash flows for a whole cohort, not for one policy. The accumulated premiums from year $j$ are $P_j\ell_{x+j}(1+i)^{k-j}$: there are $\ell_{x+j}$ policies in force when that premium is due, and each payment accumulates to time $k$. The accumulated claims from that year are $B_{j+1}d_{x+j}(1+i)^{k-j-1}$: there are $d_{x+j}$ deaths, and each claim accumulates from time $j+1$ to time $k$.

Adding the premium totals and subtracting the claim totals gives the accumulated fund for the original cohort, attributable to the policies surviving at time $k$. Dividing by $\ell_{x+k}$ gives the average amount per survivor. The life-table radix does not affect the result: multiplying every life-table count by the same constant multiplies both the cohort fund and the survivor count by that constant, leaving their ratio unchanged.

Identify the valuation duration.
Use $k$ years of past cash flows, and use $\ell_{x+k}$ as the number of policies in force at the valuation date.

Weight each premium by survivors.
For a premium due at time $j$, use $\ell_{x+j}$, then accumulate it for $k-j$ years.

Weight each claim by deaths.
For a benefit paid at time $j+1$, use $d_{x+j}$, then accumulate it for $k-j-1$ years.

Net and divide.
Subtract accumulated claims from accumulated premiums, then divide by $\ell_{x+k}$ to obtain the reserve per survivor.

Worked Applications

Worked Example: A Level Premium and a First-Year Reserve

A hypothetical two-year term contract has a death benefit of $1,000 payable at the end of the year of death. The life-table counts are $\ell_x=1{,}000$, $\ell_{x+1}=900$, and $\ell_{x+2}=720$. Premiums are due at the start of each year while the policy is in force. The effective annual interest rate is $i=10\%$. First calculate the net level premium by the equivalence principle, then calculate the retrospective reserve per survivor at duration 1.

Method and premium. The deaths in the two years are $d_x=1{,}000-900=100$ and $d_{x+1}=900-720=180$. Let $v=1/1.10$. Per policy at issue, the expected present value of benefits is

100v+180v^2 = 100\left(\frac{10}{11}\right)+180\left(\frac{100}{121}\right) = \frac{29{,}000}{121}.

The expected present value of premiums per policy at issue is $P(1+0.9v)$: all 1,000 policies pay at time 0, and 900 survive to pay at time 1. Therefore,

P\left(1+0.9\frac{10}{11}\right)=\frac{29{,}000}{121}, \qquad P\left(\frac{20}{11}\right)=\frac{29{,}000}{121}, \qquad P=\frac{1{,}450}{11}=\$131.81818\ldots.

Reserve at duration 1. By time 1, the time-0 premiums have accumulated for one year. The 100 first-year claims are paid at time 1, so they accumulate for zero years. The cohort fund at time 1 is

1{,}000P(1.10)-100(1{,}000) = 1{,}000\left(\frac{1{,}450}{11}\right)(1.10)-100{,}000 = 145{,}000-100{,}000 = \$45{,}000.

There are 900 survivors, so the reserve per survivor is $45{,}000/900=\boxed{\$50.00}$. As a check, the expected value at time 1 of the remaining one-year benefit per survivor is $0.2(1{,}000)/1.10=\$181.82$, while the premium due at time 1 is $131.81818…; their difference is $50.00. The calculation avoids the mistake of dividing the cohort fund by the original 1,000 policies instead of the 900 survivors.

Worked Example: Varying Premiums and Benefits

A hypothetical three-year policy has life-table counts $\ell_x=1{,}200$, $\ell_{x+1}=1{,}080$, $\ell_{x+2}=900$, and $\ell_{x+3}=720$. The premium due at times 0, 1, and 2 is, respectively, $150, $180, and $200. The death benefits paid at the ends of years 1, 2, and 3 are, respectively, $500, $600, and $700. Calculate the reserve per survivor at duration 3, using $i=5\%$.

Method. First find the deaths: $d_x=1{,}200-1{,}080=120$, $d_{x+1}=1{,}080-900=180$, and $d_{x+2}=900-720=180$. Accumulate each premium and claim to time 3.

The accumulated premiums are

\begin{aligned} 150(1{,}200)(1.05)^3 &= \$208{,}372.50,\\ 180(1{,}080)(1.05)^2 &= \$214{,}326.00,\\ 200(900)(1.05) &= \$189{,}000.00. \end{aligned}

Their total is $\$208{,}372.50+\$214{,}326.00+\$189{,}000.00=\$611{,}698.50$. The accumulated claims are

\begin{aligned} 500(120)(1.05)^2 &= \$66{,}150.00,\\ 600(180)(1.05) &= \$113{,}400.00,\\ 700(180) &= \$126{,}000.00. \end{aligned}

Their total is $\$305{,}550.00$. Thus the accumulated cohort fund is $\$611{,}698.50-\$305{,}550.00=\$306{,}148.50$. Dividing by the 720 policies surviving to duration 3 gives

{}_3V_x=\frac{306{,}148.50}{720}=\$425.20625\approx\boxed{\$425.21}.

The result is rounded to the nearest cent. The calculation avoids treating all premiums as if they were paid at issue or all claims as if they were paid at the same time.

Worked Example: Solving for a Premium From a Target Reserve

A hypothetical two-year contract has $\ell_x=500$, $\ell_{x+1}=450$, and $\ell_{x+2}=360$. A $1,000 death benefit is paid at the end of the year of death. A level premium $P$ is due at the start of each year while the policy remains in force. If $i=5\%$, what premium produces a retrospective reserve of $100 per survivor at duration 2?

Method. The death counts are $d_x=500-450=50$ and $d_{x+1}=450-360=90$. At time 2, the two premium totals are $500P(1.05)^2=551.25P$ and $450P(1.05)=472.5P$. The two claim totals are $50(1{,}000)(1.05)=\$52{,}500$ and $90(1{,}000)=\$90{,}000$. The target reserve means the cohort fund must be $360(\$100)=\$36{,}000$. Set accumulated premiums less accumulated claims equal to that target:

(551.25+472.5)P-(52{,}500+90{,}000)=36{,}000, \qquad 1{,}023.75P=178{,}500, \qquad P=\frac{178{,}500}{1{,}023.75}=\$174.35897\ldots.

The premium is approximately $\boxed{\$174.36}$ to the nearest cent. Checking with the unrounded premium gives a cohort fund of $\$36{,}000$, and $\$36{,}000/360=\$100$ per survivor. Using the rounded premium gives a reserve of approximately $100.00 to the nearest cent.

Common Timing and Denominator Traps

The most frequent errors come from using the right counts at the wrong times. Premiums due at time $j$ are weighted by $\ell_{x+j}$, not by $d_{x+j}$. Claims paid at time $j+1$ are weighted by $d_{x+j}$, not by the number alive at the start of that year. Also, the final division is by the number surviving at the valuation date, not the initial cohort size.

For example, in the first worked example, using the number alive at the start of year 1 as the denominator would give $45{,}000/1{,}000=\$45$, which is not the reserve per policy in force at duration 1. The correct denominator is $\ell_{x+1}=900$, giving $45{,}000/900=\$50$. The $45 figure is a cohort fund divided by the original issue count, a different quantity.

A negative result is mathematically possible if accumulated past premiums are less than accumulated past claims. It indicates a deficit under the specified retrospective calculation; it should not be silently replaced by zero unless the contract or question explicitly imposes such a rule. Finally, when the life-table radix is rescaled, keep the numerator and denominator on the same scale.

Key takeaway: For each past policy year, count premiums with the survivors at the premium due date and claims with the deaths in the year. Accumulate each cash flow from its payment time to the valuation date, net the cohort totals, and divide by the survivors at that date.

Check Your Understanding

For each problem, apply the stated annual timing and round monetary answers to the nearest cent unless an exact value is requested.

A one-year contract has $\ell_x=800$, $\ell_{x+1}=720$, a premium of $100 due at time 0, and a $1,000 death benefit due at time 1. If $i=0$, calculate the reserve per survivor at duration 1. (Answer: $0.00)
A two-year contract has $\ell_x=1{,}000$, $\ell_{x+1}=900$, and $\ell_{x+2}=750$. A level premium of $120 is due at the start of each year in force, and a $1,000 benefit is paid at the end of the year of death. If $i=10\%$, calculate the reserve per survivor at duration 2. (Answer: $5.33)
A two-year contract has $\ell_x=600$, $\ell_{x+1}=540$, and $\ell_{x+2}=420$. Premiums at times 0 and 1 are $80 and $100. Death benefits at times 1 and 2 are $500 and $800. Calculate the reserve per survivor at duration 2 when $i=4\%$. (Answer: $-\$45.53$)
A one-year contract has $\ell_x=1{,}000$, $\ell_{x+1}=920$, a premium of $150 due at time 0, and an $1,000 death benefit due at time 1. If $i=5\%$, calculate the reserve per survivor at duration 1. (Answer: $84.24)
A two-year contract has $\ell_x=1{,}000$, $\ell_{x+1}=900$, and $\ell_{x+2}=800$. The death benefit is $500 at the end of either year, and a level premium $P$ is due at the start of each year in force. If $i=0$, what premium gives a reserve of $50 per survivor at duration 2? (Answer: $P=140{,}000/1{,}900\approx \$73.68$)

What You'll Learn