Whole-Life Reserves with Annual Level Premiums

Why Premium Timing Matters in a Whole-Life Reserve

A whole-life reserve at duration $t$ measures the value, at that time, of the contract’s future benefits less the value of its future premiums. The insured’s attained age matters because the remaining death benefit and premium stream are valued using survival from that age onward. So does the instant at which the reserve is measured: a premium due at time $t$ is still outstanding just before it is paid, but is no longer outstanding just after it is paid.

This tutorial uses a discrete whole-life insurance with a level annual premium payable in advance while the insured is alive. The death benefit is paid at the end of the year of death. Unless stated otherwise, reserves are measured immediately before the premium due at duration $t$. The formulas use a net premium basis: expenses and profit loadings are not included.

Definition: For a whole-life policy with death benefit $B$ and annual premium $P$, the prospective reserve at duration $t$, immediately before the premium due at $t$, is the expected present value of future benefits minus the expected present value of future premiums, conditional on the insured being alive at $t$. With unit-benefit whole-life insurance value $A_{x+t}$ and whole-life annuity-due value $\ddot{a}_{x+t}$, this is $$ {}_tV_x = B A_{x+t} - P\ddot{a}_{x+t}. $$

Here $x+t$ is the attained age at duration $t$. The annuity-due value includes a payment immediately at the valuation time, followed by payments at later annual dates only if the insured is alive then. That first payment is why $\ddot{a}_{x+t}$, rather than $a_{x+t}$, applies immediately before a premium date. Immediately after that premium has been paid, the future premium stream starts one year later, and its value is $P a_{x+t}$.

Level Net Premium and the Reserve Formula

The level annual net premium is set by the equivalence principle at issue: the expected present value of premiums equals the expected present value of benefits. For a unit death benefit, this gives $P=A_x/\ddot{a}_x$. For a benefit of $B$, the corresponding premium is $P=B A_x/\ddot{a}_x$. The premium is calculated at issue; at duration $t$, the reserve uses the values for the attained age $x+t$.

P = \frac{B A_x}{\ddot{a}_x}, \qquad {}_tV_x = B A_{x+t} - P\ddot{a}_{x+t} \quad\text{(immediately before the premium at }t\text{).}

The insurance value $A_{x+t}$ is for a unit benefit paid at the end of the year of death. The annuity value $\ddot{a}_{x+t}$ is for unit annual payments in advance while the insured is alive. Both values use the same interest and mortality basis. If supplied values are for a unit benefit, multiply the insurance value by the actual benefit amount; do not multiply the annuity value by the death benefit unless the premium itself scales with that benefit.

Timing result: Immediately after the premium at duration $t$ has been paid, the reserve is $$ {}_tV_x^{\text{after}} = B A_{x+t} - P a_{x+t}, \qquad a_{x+t}=\ddot{a}_{x+t}-1. $$ The before-premium reserve exceeds the after-premium reserve by $P$, because the premium due at $t$ is included in the former’s future premiums but not the latter’s.

The relation between the whole-life insurance and annuity values can also help check factors. Let $K$ be the curtate future lifetime at age $x$, let $v=1/(1+i)$, and let $d=1-v$. A unit benefit paid at the end of the year of death has present value $v^{K+1}$. The present value of a unit annuity-due stream paid while alive is $1+v+\cdots+v^K=(1-v^{K+1})/d$. Taking expectations gives the standard identity below.

A_x = E[v^{K+1}], \qquad \ddot{a}_x = E\!\left[\frac{1-v^{K+1}}{d}\right], \qquad A_x+d\ddot{a}_x=1.

This identity is useful as a factor check, provided the insurance and annuity values use the same mortality and interest assumptions. It does not replace the prospective reserve calculation: at duration $t$, use the values at age $x+t$, and subtract only premiums that remain outstanding under the stated timing convention.

Worked Applications

Worked Example: Reserve Before a Premium Date

A hypothetical whole-life policy pays \$1 at the end of the year of death. Interest is $i=5\%$, so $v=1/1.05=20/21$ and $d=1/21$. On the hypothetical mortality basis, $A_{x}=0.6000$ and $A_{x+1}=0.5500$. Find the annual net premium and the reserve at duration 1 immediately before the premium due then.

Method. Calculate the issue annuity value from the insurance-annuity identity, use equivalence to find $P$, then value the remaining benefit and premiums at age $x+1$. The assumed values are coherent with a one-year death probability $q_x=8/45$: $v[q_x+(1-q_x)A_{x+1}]=(20/21)[8/45+(37/45)(0.55)]=0.6000$.

At issue, the annuity-due value is $\ddot{a}_x=(1-A_x)/d=0.4000/(1/21)=8.4000$. Therefore the level annual net premium is $P=A_x/\ddot{a}_x=0.6000/8.4000=1/14=0.0714286$.

At duration 1, $A_{x+1}=0.5500$, and $\ddot{a}_{x+1}=(1-0.5500)/(1/21)=9.4500$. Thus ${}_1V_x=0.5500-(1/14)(9.4500)=0.5500-0.6750=-0.1250$. The reserve is $-\$0.13$ per dollar of benefit, rounded to the nearest cent.

Check and trap avoided. At issue, $P\ddot{a}_x=(1/14)(8.4)=0.6000=A_x$, as required by equivalence. A negative reserve is possible: surviving to the attained age can leave expected future premiums, on this basis, with a greater value than the remaining benefit. Do not replace a negative result with zero.

Worked Example: Before Versus After the Premium

A hypothetical unit-benefit whole-life policy has $i=4\%$. On its mortality basis, $A_x=0.5000$ and $A_{x+1}=0.4500$. Find the issue net premium and the reserve at duration 1 both immediately before and immediately after the annual premium is paid.

Method. First find the annuity-due values using $d=1-v=1-1/1.04=1/26$. Then use the appropriate premium annuity at each valuation instant. The one-year mortality probability is $q_x=7/55$, since $(1/1.04)[7/55+(48/55)(0.45)]=0.5000$.

At issue, $\ddot{a}_x=(1-0.5000)/(1/26)=13.0000$, so $P=0.5000/13=1/26=0.0384615$. At age $x+1$, $\ddot{a}_{x+1}=(1-0.4500)/(1/26)=14.3000$, hence $a_{x+1}=14.3000-1=13.3000$.

Immediately before the premium at duration 1, ${}_1V_x=0.4500-(1/26)(14.3000)=0.4500-0.5500=-0.1000$. Immediately after it, ${}_1V_x^{\text{after}}=0.4500-(1/26)(13.3000)=0.4500-0.5115385=-0.0615385$. Rounded to four decimal places, the reserves are $-0.1000$ and $-0.0615$ per dollar of benefit, respectively. Their difference is $0.0384615=P$, as expected.

Check and trap avoided. Using the annuity-due value after the premium incorrectly counts the payment just made as still outstanding. Conversely, using the annuity-immediate value just before payment omits the premium due at that instant.

Worked Example: Scaling a Reserve to the Contract Benefit

A hypothetical whole-life policy pays \$120,000 at the end of the year of death and charges a level annual net premium in advance while the insured is alive. The interest rate is $5\%$. On the hypothetical mortality basis, $A_x=0.6500$ at issue and $A_{x+5}=0.7000$ at duration 5. Find the annual premium and the reserve immediately before the premium at duration 5.

Method. Convert each unit-benefit insurance value to its annuity-due value, calculate the benefit-scaled premium at issue, and then value the remaining contract at age $x+5$.

Here $d=1/21$. At issue, $\ddot{a}_x=(1-0.6500)/(1/21)=7.3500$. The annual net premium is $P=120{,}000(0.6500)/7.3500=\$10{,}612.2449$, or $\$10{,}612.24$ to the nearest cent.

At duration 5, $\ddot{a}_{x+5}=(1-0.7000)/(1/21)=6.3000$. The value of the remaining benefit is $120{,}000(0.7000)=\$84{,}000$. The value of the future premiums is $\$10{,}612.2449(6.3000)=\$66{,}857.1429$. Therefore ${}_5V_x=\$84{,}000-\$66{,}857.1429=\$17{,}142.8571$, or $\$17{,}142.86$ to the nearest cent.

Check and trap avoided. The premium is determined from issue-age values, whereas the reserve uses duration-5 values. The attained-age insurance value is multiplied by \$120,000; the annuity value is multiplied by the annual premium, not by the benefit.

Common Errors and Practical Checks

A reserve is a conditional value at the specified duration, not the issue-date value carried forward without updating the mortality basis. Before substituting values, identify the attained age and decide whether the valuation instant is before or after the premium due at that date. Then make sure the premium and benefit terms use consistent units and that the actuarial values share the same interest and mortality basis.

Using the wrong premium timing. Before a premium date, include the payment due then and use $\ddot{a}_{x+t}$. After it, omit that payment and use $a_{x+t}$.
Using issue-age factors at duration $t$. The reserve values future cash flows from age $x+t$, not age $x$.
Mixing unit and dollar amounts. If $A$ and $\ddot{a}$ are unit-benefit factors, multiply $A$ by the death benefit and the annuity factor by the annual premium.
Assuming every reserve must be positive. A negative prospective reserve can result when the value of future premiums exceeds the value of future benefits.

Key takeaway: State the reserve timing first. Immediately before the premium at duration $t$, subtract $P\ddot{a}_{x+t}$ from $B A_{x+t}$; immediately after it, subtract $P a_{x+t}$. Calculate the level net premium from issue-age values, and value the remaining contract at the attained age.

Check Your Understanding

For each problem, use the stated timing convention and keep full precision until the final answer.

A unit-benefit whole-life policy has $i=5\%$, $A_x=0.4800$, and $A_{x+2}=0.6000$ on a hypothetical mortality basis. Find the annual net premium and the reserve at duration 2 immediately before the premium. (Answer: $P=4/91=0.043956$; reserve $=0.2308$ per dollar of benefit.)
At $i=4\%$, a hypothetical policy has $A_x=0.5400$ and $A_{x+1}=0.5000$. Find the unit-benefit net premium, then calculate the reserve at duration 1 immediately after the premium. (Answer: $P=0.5400/11.9600=0.04515$; reserve $=0.5000-P(12.0000)=-0.0418$ per dollar of benefit.)
A \$50,000 whole-life benefit has an annual net premium of \$2,000. At duration 8 immediately before the premium, the unit-benefit factors are $A_{x+8}=0.6200$ and $\ddot{a}_{x+8}=7.6000$. Find the reserve. (Answer: $\$31{,}000-\$15{,}200=\$15{,}800.00$.)
A unit-benefit policy has $i=5\%$, $A_x=0.5500$, and $A_{x+4}=0.6500$. Find the annual net premium and the reserve at duration 4 immediately after the premium, using $a_{x+4}=\ddot{a}_{x+4}-1$. (Answer: $\ddot{a}_x=9.4500$, $P=0.05820$, $\ddot{a}_{x+4}=7.3500$, and reserve $=0.6500-P(6.3500)=0.2804$ per dollar of benefit.)

What You'll Learn