Calculating Reserves from Life-Table Columns

Why Life-Table Columns Matter

A prospective reserve compares the expected present value of benefits still owed with the expected present value of premiums still due. When mortality is supplied as a life table, the key calculation is to convert its survivor counts into probabilities conditional on the insured being alive at the valuation date. Using counts from the table without that conditioning can materially distort the reserve.

This tutorial works at integer policy durations and annual payment dates. The valuation convention is important: reserves are calculated immediately before any premium due at that date. Thus, if the insured is alive at duration $k$, a premium due at duration $k$ is included among future premiums. Death benefits are paid at the end of the year of death. All examples use a deterministic effective annual interest rate $i$, a stated life table, and no expenses, lapses, or other policy cash flows.

Definition: A life table gives $\ell_y$, the number of lives surviving to exact age $y$ in a hypothetical cohort. For an insured aged $y$, with $\ell_y>0$, the conditional probability of surviving $j$ years is ${}_jp_y=\ell_{y+j}/\ell_y$. The conditional probability of dying during the year from age $y+j$ to $y+j+1$, given survival to age $y$, is $(\ell_{y+j}-\ell_{y+j+1})/\ell_y$.

From Survivor Counts to Conditional Probabilities

The denominator in each probability is the number alive at the valuation age, not necessarily the number alive at policy issue. For example, among $\ell_y$ people alive at age $y$, $\ell_{y+1}$ survive one year. Therefore the one-year conditional survival probability is $\ell_{y+1}/\ell_y$, and the conditional death probability is $(\ell_y-\ell_{y+1})/\ell_y$. For later years, compare the relevant life-table column with the same starting count $\ell_y$.

For a reserve at duration $k$ on a policy issued at age $x$, the valuation age is $y=x+k$. A death in the $j$th future policy year means death between ages $x+k+j-1$ and $x+k+j$. Its probability, conditional on being alive at valuation, is the difference between those two survivor counts divided by $\ell_{x+k}$.

P(\text{death in future year }j\mid\text{alive at }x+k) = \frac{\ell_{x+k+j-1}-\ell_{x+k+j}}{\ell_{x+k}}, \qquad j=1,2,\ldots

Likewise, the chance of being alive at the start of future policy year $j+1$ is $\ell_{x+k+j}/\ell_{x+k}$. This is the relevant probability for a premium due at that date. A premium due immediately at the valuation date has probability 1, because the reserve is conditional on the insured being alive then.

These expressions assume the life-table columns describe the mortality model for the policy being valued, with no unmodelled selection, lapses, or changes in coverage. The life-table radix—the cohort size at some reference age—cancels in the ratios, so the table need not start with a particular number of lives. If $\ell_{x+k}=0$, conditioning on survival to that age is not defined, and the reserve cannot be calculated by these ratios.

Prospective Reserve Formulas from the Columns

Consider an $n$-year term policy with benefit $B$ payable at the end of the year of death and level annual premium $P$. At duration $k$, suppose $n$ years of coverage remain and premiums are due at the valuation date and each subsequent policy anniversary through the last covered year. With $v=1/(1+i)$, the prospective reserve is the present value of remaining benefits less the present value of remaining premiums.

{}_kV_x = B\sum_{j=1}^{n}v^j \frac{\ell_{x+k+j-1}-\ell_{x+k+j}}{\ell_{x+k}} - P\sum_{j=0}^{n-1}v^j \frac{\ell_{x+k+j}}{\ell_{x+k}}.

The benefit in future year $j$ is discounted $j$ years because it is paid at that year’s end. Premium $j$ is due at the start of that future year and is discounted $j$ years. The first premium term is $P$, without a survival adjustment, because it is due immediately at valuation.

If premiums stop before coverage ends, keep all remaining benefit terms but include only premium dates still covered by the premium-paying period. For an endowment with death benefit $B$ during the remaining $n$ years and maturity benefit $M$ payable at the end of year $n$ if the insured survives, add the conditional present value of the maturity payment:

M v^n\frac{\ell_{x+k+n}}{\ell_{x+k}}.

When $M=B$, death and maturity together pay the same amount at the end of the remaining term, whether death occurs during that year or the insured survives to maturity. The probability terms should still be constructed from the life table, rather than treating the maturity payment as a death benefit.

Worked Calculations

Worked Example: A Term Reserve from Survivor Counts

A hypothetical three-year term policy is issued at age $x$, with a death benefit of $10,000 payable at each year’s end of death and annual premiums of $900. At duration 1, calculate the reserve immediately before the premium due then. The effective annual interest rate is $i=5\%$, and the relevant life-table columns are $\ell_{x+1}=900$, $\ell_{x+2}=810$, and $\ell_{x+3}=729$.

Method and timing. Two coverage years remain, and two premiums remain: one due now and one due one year later if the insured survives. Set $v=1/1.05=20/21$.

Conditional death probabilities. From valuation age $x+1$, death in the first remaining year has probability $(900-810)/900=0.10$. Death in the second has probability $(810-729)/900=0.09$. Survival to the end of the second year is $729/900=0.81$. These outcomes sum to $0.10+0.09+0.81=1$.

Benefit value. Discount the first-year death benefit one year and the second-year benefit two years:

10{,}000\left(0.10v+0.09v^2\right) = 10{,}000\left(0.10\cdot\frac{20}{21} +0.09\cdot\frac{400}{441}\right) = \$1{,}768.71.

Premium value. The premium due now is certain conditional on being alive at valuation. The later premium is due if the insured survives one year:

900\left(1+v\cdot\frac{810}{900}\right) = 900\left(1+\frac{20}{21}\cdot0.90\right) = \$1{,}671.43.

The reserve is $1{,}768.71-1{,}671.43=\boxed{\$97.28}$, rounded to the nearest cent. The method avoids using the issue-age count as the denominator: the correct conditioning count is $\ell_{x+1}=900$. The positive reserve is reasonable because the discounted remaining benefit value exceeds the remaining premium value.

Worked Example: Coverage Lasts Longer Than Premium Payments

A hypothetical four-year term policy has a $20,000 end-of-year death benefit and annual premiums of $1,800 payable only at durations 0 and 1. Calculate the reserve at duration 1, immediately before that year’s premium. The effective annual interest rate is $i=4\%$. The life-table columns are $\ell_{x+1}=920$, $\ell_{x+2}=846.4$, $\ell_{x+3}=778.688$, and $\ell_{x+4}=716.39296$.

Method and remaining periods. Three years of insurance coverage remain, but only one premium remains: the premium due immediately at duration 1. Set $v=1/1.04=25/26$. The table implies a one-year survival ratio of $0.92$ at each of the remaining ages.

Benefit probabilities. Conditional on being alive at duration 1, the death probabilities in the next three years are $0.08$, $0.92(0.08)=0.0736$, and $0.92^2(0.08)=0.067712$. Thus the benefit value is:

20{,}000\left(0.08v+0.0736v^2+0.067712v^3\right) = 20{,}000\left[\frac{1}{13} \left(1+\frac{23}{26}+\left(\frac{23}{26}\right)^2\right)\right] = \$4{,}103.32.

Here $0.08v=1/13$, and the ratio between successive discounted death-probability terms is $0.92v=23/26$. The expression evaluates to a benefit present value of $4,103.32.

Premium value and reserve. The only remaining premium is due immediately, so its value is $1,800. Therefore ${}_1V_x=4{,}103.32-1{,}800=\boxed{\$2{,}303.32}$, rounded to the nearest cent. The key mistake avoided is stopping the benefit calculation when premiums stop: coverage continues for three more years, so all three death-benefit terms belong in the reserve.

Worked Example: An Endowment Reserve Including Maturity

A hypothetical three-year endowment policy pays $10,000 at the end of the year of death or, if the insured survives, at maturity. Annual premiums of $2,500 are due at durations 0, 1, and 2. Calculate the reserve at duration 1 immediately before the premium due then. Use $i=5\%$ and the life-table columns $\ell_{x+1}=900$, $\ell_{x+2}=810$, and $\ell_{x+3}=729$.

Method. Two years remain. Conditional on being alive at duration 1, death in the first remaining year has probability $90/900=0.10$, death in the second has probability $81/900=0.09$, and survival to maturity has probability $729/900=0.81$. These probabilities total 1. Set $v=20/21$.

Benefit value. The two death benefits are discounted to the ends of their respective years. The maturity payment is discounted two years and weighted by the probability of survival:

10{,}000\left(0.10v+0.09v^2+0.81v^2\right) = 10{,}000\left(\frac{2}{21}+\frac{360}{441}\right) = \$9{,}115.65.

Premium value. There is a premium due immediately and another due one year later if the insured survives:

2{,}500\left(1+0.90v\right) = 2{,}500\left(1+\frac{18}{21}\right) = \$4{,}642.86.

Subtracting gives ${}_1V_x=9{,}115.65-4{,}642.86=\boxed{\$4{,}472.79}$, rounded to the nearest cent. As a check, the benefit probabilities are $0.10+0.09+0.81=1$, and the maturity payment has been included only for survival to maturity. The calculation avoids omitting the endowment payment or counting it again as a death benefit.

Derivation and Checks

The life-table reserve formula follows directly from conditional expected present values. At valuation age $y=x+k$, there are $\ell_y$ survivors in the table. Of these, $\ell_{y+j-1}-\ell_{y+j}$ die during future year $j$. Conditional on being alive at valuation, the probability of that event is therefore the number who die divided by the number alive at valuation. Multiplying the benefit by that probability and discounting it $j$ years gives its expected present value.

For a premium due $j$ years after valuation, the relevant count is instead the number alive at that payment date, $\ell_{y+j}$. Its conditional payment probability is $\ell_{y+j}/\ell_y$. Discounting the premium by $v^j$, then summing the expected present values, gives the premium side of the prospective reserve. This also explains why a premium due immediately has probability $1$, while later premiums are survival-weighted.

Several checks help catch errors before finalizing a result:

For a term ending after $n$ years, the conditional death probabilities over those years plus the probability of surviving all $n$ years must sum to 1.
Use the survivor count at the valuation age as the denominator for every remaining probability.
Discount a benefit paid at year-end by the number of years to that year-end; discount a premium at the beginning of a future year by the number of years to that payment date.
Stop premium terms when the premium-paying period ends, not when the insurance coverage ends.
Apply the reserve definition consistently as benefit value minus premium value. A negative reserve can occur for some cash-flow patterns and is not, by itself, evidence of an arithmetic error.

Key takeaway: At an integer valuation age, divide relevant future survivor counts or deaths by the survivor count at valuation. Then discount each possible benefit and premium to the valuation date, respecting its payment time and the period for which it is payable.

Check Your Understanding

For each problem, use the stated valuation convention: immediately before any premium due at the valuation date. Benefits are paid at the end of the year of death unless a maturity payment is specified.

A two-year term policy is valued at issue. It has a $8,000 death benefit, two annual premiums of $700, and $i=0\%$. The table has $\ell_{60}=5{,}000$, $\ell_{61}=4{,}500$, and $\ell_{62}=4{,}050$. Calculate the reserve. (Answer: $190.00)
A two-year term policy is valued at age 71 with two years of coverage and premiums remaining. The death benefit is $6,000, the annual premium is $500, and $i=5\%$. Use $\ell_{71}=900$, $\ell_{72}=810$, and $\ell_{73}=729$. Calculate the reserve. (Answer: $132.65)
A two-year endowment is valued at age 80 before the first of two remaining premiums. It pays $12,000 on death during the term or on survival to maturity; the annual premium is $4,000 and $i=4\%$. The table has $\ell_{80}=1{,}000$, $\ell_{81}=950$, and $\ell_{82}=900$. Calculate the reserve. (Answer: $3,463.02)
A three-year term policy has a $5,000 death benefit and premiums of $700 due only at durations 0 and 1. At duration 1, immediately before the premium, calculate the reserve using $i=0\%$ and $\ell_{40}=1{,}000$, $\ell_{41}=900$, $\ell_{42}=810$, $\ell_{43}=729$. (Answer: $250.00)
A two-year endowment pays $10,000 at death during the term or at maturity. Find its level annual net premium at issue, then its reserve at duration 1 immediately before the second premium. Use $i=5\%$ and $\ell_{30}=1{,}000$, $\ell_{31}=900$, $\ell_{32}=810$. (Answer: net premium $4,908.42; reserve $4,615.38)

What You'll Learn