Limited-Payment Whole-Life Reserves

Why Premium-Payment Duration Matters

A whole-life policy can provide coverage for the insured’s entire future lifetime even when premiums are payable for only a limited number of years. Once the scheduled premium period ends, the death benefit remains in force, but the future premium stream has disappeared. A reserve calculation must reflect both parts of that change: the benefit is still valued as whole-life insurance, while the value of future premiums depends on how many remain.

Timing matters as well. At a premium date, a reserve measured immediately before the payment includes that payment among future premiums. Immediately after the payment, it does not. The difference between these two reserves is the premium just paid.

Definition: Consider a discrete whole-life policy issued at age $x$, with benefit $B$ payable at the end of the year of death. Level premiums $P$ are payable annually in advance while the insured is alive, for at most $m$ years: at times $0,1,\ldots,m-1$. In this tutorial, ${}_kV_x$ denotes the prospective reserve at duration $k$, immediately before any premium due at time $k$. The reserve immediately after that payment is denoted ${}_kV_x^+$. At $k=m$, no premium is due.

Values Needed for the Calculation

The whole-life insurance value $A_{x+k}$ is the expected present value, at age $x+k$, of a unit benefit paid at the end of the year of death. A benefit of $B$ therefore has value $B A_{x+k}$. For premiums, define the temporary life annuity-due value by

\ddot{a}_{x:\overline{n}|} = \sum_{j=0}^{n-1} v^j\,{}_jp_x,

where $n$ is the number of possible annual payments, the first is due immediately, and each later payment is made only if the insured is alive at its due date. In particular, this value is for premiums paid for at most $n$ years, not for coverage lasting $n$ years.

If $P$ is a net level premium determined by the equivalence principle at issue, the present value of premiums equals the present value of benefits:

P\ddot{a}_{x:\overline{m}|}=B A_x, \qquad P=\frac{B A_x}{\ddot{a}_{x:\overline{m}|}}.

The reserve formulas below also apply when $P$ is specified rather than calculated by this equation. In that case, the reserve measures the prospective difference using that specified premium.

Prospective Reserves Before and After a Premium

At duration $k<m$, immediately before the premium due at time $k$, the insured has attained age $x+k$. There are $m-k$ scheduled premium dates remaining, including the current date. Once the last premium date has passed, there are no future premiums; the death benefit still has whole-life value.

{}_kV_x = \begin{cases} B A_{x+k}-P\ddot{a}_{x+k:\overline{m-k}|}, & 0\leq k<m,\\[4pt] B A_{x+k}, & k\geq m. \end{cases}

Immediately after paying the premium at $k<m$, only the remaining $m-k-1$ scheduled premiums are future payments. Thus

{}_kV_x^+ = B A_{x+k}-P\ddot{a}_{x+k:\overline{m-k-1}|},

where an annuity with zero possible payments has value zero. Subtracting the two prospective expressions gives ${}_kV_x^+={}_kV_x+P$. The post-payment reserve is higher because the premium has been removed from the future outgo.

Key result: At the final premium date $m-1$, the reserve immediately before payment is $B A_{x+m-1}-P$. Immediately after that premium, it is $B A_{x+m-1}$. At duration $m$, the reserve is $B A_{x+m}$: no premiums remain, but whole-life coverage continues.

Why the Formula Holds

The prospective reserve is the conditional expected present value of future benefits minus the conditional expected present value of future premiums, given that the insured is alive at the valuation time. At duration $k<m$, the benefit is a whole-life benefit at attained age $x+k$, so its value is $B A_{x+k}$. The scheduled payments at times $k$ through $m-1$ form an $m-k$-payment life annuity-due, so their value is $P\ddot{a}_{x+k:\overline{m-k}|}$. Their difference is the stated reserve. For $k\geq m$, the premium value is zero, leaving only the whole-life benefit value.

This argument also explains why premium cessation does not turn the policy into term insurance or endowment insurance. The duration of premium payments restricts the premium stream, not the duration of the insured benefit.

Worked Example: A Two-Payment Policy at the Second Premium Date

A hypothetical policy pays a benefit of 1 at the end of the year of death. It charges level net premiums at times 0 and 1, while the insured is alive. Assume an effective annual interest rate $i=25\%$, so $v=0.8$. Assume a geometric mortality model with ${}_1q_{x+k}=0.2$ at every future age $x+k$, and hence ${}_1p_{x+k}=0.8$. Calculate the net premium and the reserves immediately before and after the time-1 premium.

Method. First calculate the whole-life insurance and annuity values under the stated model, then apply the prospective formulas. This keeps the premium timing separate from the lifetime of the benefit.

For a unit whole-life benefit, the probability of death in year $j+1$ is $0.8^j(0.2)$, and the payment is discounted $j+1$ years. Therefore

A_x=\sum_{j=0}^{\infty}(0.8)^j(0.2)(0.8)^{j+1} =0.16\sum_{j=0}^{\infty}(0.64)^j =\frac{0.16}{1-0.64} =\frac{4}{9}.

The whole-life annuity-due value is $\ddot{a}_x=\sum_{j=0}^{\infty}(0.64)^j=25/9$. For the two-payment premium period,

\ddot{a}_{x:\overline{2}|}=1+0.8(0.8)=1+0.64=\frac{41}{25}, \qquad P=\frac{4/9}{41/25}=\frac{100}{369}\approx 0.2710.

At duration 1, immediately before the second premium, one premium remains. The reserve is

{}_1V_x=\frac{4}{9}-\frac{100}{369} =\frac{164-100}{369} =\frac{64}{369}\approx 0.1734.

Immediately after that premium, no future premiums remain, so ${}_1V_x^+=A_{x+1}=4/9\approx0.4444$. The check is ${}_1V_x^+-{}_1V_x=164/369-64/369=100/369=P$. The common mistake is to subtract a premium after the final scheduled payment; doing so would incorrectly reduce the post-payment reserve.

Worked Example: Three Premiums and the Last Payment

A hypothetical policy pays 1 at the end of the year of death and charges net annual premiums at times 0, 1, and 2. Assume $i=25\%$ and geometric mortality with annual death probability $q=0.2$ at every age, as in the preceding example. Calculate the net premium, the reserve at duration 1 immediately before its premium, and the reserve at duration 2 immediately after the last premium.

Method. The benefit value is $A_x=4/9$. The three-payment annuity-due value is

\ddot{a}_{x:\overline{3}|}=1+0.64+0.64^2 =1+0.64+0.4096=2.0496=\frac{1281}{625}.

Thus the net premium per unit of benefit is

P=\frac{4/9}{1281/625} =\frac{2500}{11529} \approx 0.2168.

At duration 1 before the premium, two payments remain. Their value is $1+0.64=41/25$, so

{}_1V_x=\frac{4}{9}-\frac{2500}{11529}\frac{41}{25} =\frac{5124}{11529}-\frac{4100}{11529} =\frac{1024}{11529} \approx 0.0888.

At duration 2 immediately after the last premium, no future premiums remain. The reserve is $A_{x+2}=4/9\approx0.4444$. The reserve immediately before that last payment would instead be $4/9-P\approx0.2276$; the increase after payment is exactly $P$. Under this constant mortality model the whole-life value is the same at each attained age. With age-varying mortality, the value at age $x+2$ must be calculated at that age rather than assumed equal to $A_x$.

A One-Year Recursion Check

A prospective reserve can be checked against the cash flows over the next year. Just before a premium at time $k<m$, paying it leaves the post-payment reserve ${}_kV_x+P$. Over the year, a death produces benefit $B$ at year-end; survival leads to the next pre-premium reserve. Thus, with $q_{x+k}={}_1q_{x+k}$ and $p_{x+k}={}_1p_{x+k}$,

{}_kV_x+P = v\left(q_{x+k}B+p_{x+k}{}_{k+1}V_x\right).

This recursion assumes annual premiums are due at the start of the year and the death benefit is paid at the end of the year of death. It is not a substitute for checking timing: if the valuation is immediately after the premium, the premium is no longer included in the reserve.

Worked Example: Checking a Reserve by Recursion

Use the two-payment policy from the first example, but let the benefit be 1,000. Assume $i=25\%$, annual death probability $q=0.2$ at every age, and premiums at times 0 and 1. Verify the reserve at duration 0 immediately before the initial premium using the one-year recursion.

The benefit value is $1{,}000(4/9)$, and the net premium scales from the unit-benefit result to

P=1{,}000\left(\frac{100}{369}\right) =\frac{100{,}000}{369} \approx \$271.00.

From the first example, the duration-1 reserve immediately before the second premium is $1{,}000(64/369)=64000/369$, or about $\$173.44$. At issue, the reserve before the initial premium is zero by the net premium equivalence principle. The recursion gives

0+P = 0.8\left(0.2(1000)+0.8\frac{64000}{369}\right) = 0.8\left(200+\frac{51200}{369}\right) = 0.8\left(\frac{125000}{369}\right) = \frac{100000}{369} =P.

The calculation checks. Treating the duration-1 reserve as a post-premium value would omit the second premium from the future cash flows and would not give this recursion.

Common Timing Traps

Including too many premiums. At duration $k$, immediately before the due payment, there are $m-k$ possible payments, not $m$. The earlier payments have already occurred.
Subtracting a premium after it has been paid. Immediately after a premium date, that premium is no longer future outgo; the post-payment reserve is higher by $P$.
Ending the benefit with the premium period. A limited-payment whole-life policy retains its whole-life benefit after premiums cease. After the final payment, the reserve is a whole-life insurance value, not zero.
Mixing up the last payment date and the next duration. The final scheduled payment is at time $m-1$. At time $m$, no premium is due, and the valuation age is $x+m$.

Key takeaway: State whether the reserve is before or after the premium due at the valuation time. Before premium cessation, subtract the value of the premiums still payable; after cessation, subtract none. The whole-life benefit remains in force throughout.

Check Your Understanding

Assume each problem describes a discrete whole-life benefit payable at the end of the year of death and annual premiums payable in advance while the insured is alive. Use the stated timing when calculating reserves.

A unit-benefit policy has two net premium payments, at times 0 and 1. Assume $i=0$ and a constant annual death probability $q=0.25$. Calculate the net premium and the reserve at duration 1 immediately before the last premium. (Answer: $P=4/7$; reserve $=3/7$.)
A unit-benefit policy has three net premium payments. Assume $i=25\%$ and constant annual death probability $q=0.2$ at every age. Calculate the net premium and the reserve at duration 2 immediately before the last premium. (Answer: $P=2500/11529\approx0.2168$; reserve $=2624/11529\approx0.2276$.)
A hypothetical policy has a benefit of $\$50{,}000$, with four net annual premiums. Assume $i=5\%$ and constant annual death probability $q=0.1$ at every age. Find the reserve immediately after the fourth premium, at time 3. (Answer: $\$33{,}333.33$, since $A_x=2/3$ under these assumptions and no premiums remain after time 3.)
A hypothetical policy has a benefit of $\$10{,}000$ and two net annual premiums. Assume $i=25\%$ and constant annual death probability $q=0.2$ at every age. Find the reserve at duration 1 both immediately before and immediately after the second premium. (Answer: before, $\$1{,}734.42$; after, $\$4{,}444.44$. The difference is the second premium, approximately $\$2{,}710.03$.)

What You'll Learn