Zillmerized Reserves

Why Zillmerize a Reserve?

An insurer may incur substantial acquisition expenses when a policy is issued, while the premiums supporting the policy arrive over time. A valuation that recognizes only future benefits and level net premiums does not explicitly allow for those initial expenses. Zillmerization modifies the valuation premium to include a specified allowance for acquisition expenses. Because the modified premium is higher, the prospective reserve is lower than it would be on the unmodified basis.

The method is a valuation convention: the allowance is spread over the expected premium-paying period. It does not mean that the policyholder necessarily pays an additional premium. The modified premium is used to calculate the reserve, not automatically substituted for the contractual gross premium. The treatment here is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: In a discrete annual model, a Zillmerized valuation premium is a valuation premium $P^Z=P+z$, where $P$ is the unmodified valuation premium and $z$ is an allowance per premium for acquisition expenses. If the specified issue-date allowance is $E_0$, then $z$ is chosen so that the present value of the allowance premiums is $E_0$. The resulting prospective reserve is called a Zillmerized reserve.

Valuation Model and Timing

Use the following model unless a problem specifies a different one: interest is at a constant effective annual rate $i$; premiums are level and payable annually in advance while the policy is in force; and a death benefit is paid at the end of the year of death. Let $v=1/(1+i)$. For an $n$-year contract, let $P$ be the net valuation premium determined by the equivalence principle. The mortality probabilities may vary by year.

In the formulas below, $V_k$ is measured at time $k$, immediately before the premium due at that time. Thus a premium due at time $k$ is included among future premium payments. There is no premium after the contract expires. This timing makes the issue-date net reserve zero when the unmodified premium satisfies equivalence.

The temporary life annuity-due factor used in the calculations is $\ddot a_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k\,{}_kp_x$, the present value at issue of one unit paid at the start of each year while the insured is alive and the policy remains in force. At duration $k$, the corresponding factor is calculated using survival from time $k$ onward.

P=\frac{\operatorname{PV}_0(\text{future benefits})}{\ddot a_{x:\overline{n}|}}, \qquad z=\frac{E_0}{\ddot a_{x:\overline{n}|}}, \qquad P^Z=P+z.

The formula for $z$ assumes that the allowance is charged at every valuation premium date while the policy remains in force, on the same premium-paying pattern used in the annuity factor. If premiums are payable for a shorter period, or only in particular states, use the present value of that actual premium pattern instead. A quoted allowance per policy is an amount at issue; $z$ is an amount per premium. They are not interchangeable.

Prospective Zillmerized Reserves

A prospective reserve is the present value, at the valuation date, of future benefits less the present value of future valuation premiums. The Zillmerized reserve uses $P^Z$ in that calculation. In the simple model above, let $a_k$ denote the value at time $k$ of the remaining unit premiums, including any premium due at time $k$. Then

V_k^Z = \operatorname{PV}_k(\text{future benefits})-P^Z a_k = V_k-z a_k.

This relation follows directly by substituting $P^Z=P+z$ into the prospective definition. The benefit present value is unchanged; the premium present value increases by $z a_k$. Therefore the reserve decreases by precisely that amount. At issue, $a_0=\ddot a_{x:\overline{n}|}$, and if the unmodified premium gives $V_0=0$, then $V_0^Z=-z\ddot a_{x:\overline{n}|}=-E_0$. The identity is a check on the allowance calibration, not a guarantee that a particular valuation regulation permits a negative reported reserve.

Worked Example: Two-Year Term Insurance

A hypothetical two-year term policy pays $10,000 at the end of the year of death. Annual premiums are payable in advance while the insured is alive. Assume $i=10\%$, with $q=0.10$ in each year and $p=0.90$. A $100 acquisition-expense allowance is to be Zillmerized. Find the unmodified premium, the allowance per premium, and the Zillmerized reserves at issue and at time 1.

Method. First calculate the benefit and premium present values, then calibrate the allowance using the issue-date premium annuity. No expense amount is added directly to the benefit.

The discount factor is $v=1/1.10=10/11$. The present value of benefits per dollar of benefit is $(0.10)(10/11)+(0.90)(0.10)(100/121)=1/11+9/121=20/121$. Thus the benefit present value is $10{,}000(20/121)=\$1{,}652.8926$.

The premium annuity-due factor is $1+(0.90)(10/11)=20/11=1.8181818$. Therefore the unmodified net premium is $(10{,}000(20/121))/(20/11)=\$10{,}000/11=\$909.0909$.

The allowance per premium is $z=100/(20/11)=\$55$. Hence $P^Z=909.0909+55=\$964.0909$.

At issue, the Zillmerized reserve is the benefit present value less the modified premium annuity: $1{,}652.8926-964.0909(20/11)=-\$100.00$. At time 1, the remaining benefit present value is $10{,}000(0.10)(10/11)=\$909.0909$, and one premium of $964.0909 is due. Thus $V_1^Z=909.0909-964.0909=-\$55.00$. At expiry there are no future benefits or premiums, so the reserve is zero.

Check. The reduction from the unmodified reserve is $100 at issue and $55 at time 1, exactly the allowance per premium multiplied by the remaining premium factors, $55(20/11)=100$ and $55(1)=55$. Amounts are rounded to the nearest cent only at the end.

Duration-by-Duration Calculation

At a later duration, the initial allowance is not generally subtracted in full. The amount reflected in the reserve is the allowance per premium multiplied by the present value of the remaining premiums on the valuation basis. This is why the remaining premium pattern and the reserve timing must be identified before calculating a Zillmerized reserve.

Worked Example: Varying Mortality in a Two-Year Contract

A hypothetical two-year policy pays $10,000 at the end of the year of death. Premiums are annual in advance. Interest is $i=0$, with first-year $q_1=0.10$ and second-year $q_2=0.20$. The specified issue-date acquisition allowance is $190. Calculate the net premium, the Zillmer premium, and the reserve at time 1.

Method. Use the survival probability from issue to weight the second-year benefit and premium. At time 1, value only the remaining year, conditional on being in force at that time.

The first-year benefit has probability $0.10$. The second-year benefit has probability $(0.90)(0.20)=0.18$. With zero interest, the issue-date benefit present value is $10{,}000(0.10+0.18)=\$2{,}800$.

The premium annuity factor is $1+0.90=1.90$. Therefore $P=2{,}800/1.90=\$1{,}473.6842$. The allowance per premium is $z=190/1.90=\$100$, giving $P^Z=\$1{,}573.6842$.

At time 1, the remaining death benefit has expected value $0.20(10{,}000)=\$2{,}000$, and one premium is due. Therefore $V_1^Z=2{,}000-1{,}573.6842=\$426.3158$. The unmodified reserve is $2{,}000-1{,}473.6842=\$526.3158$, so the Zillmerization reduces it by $100, the allowance on the one remaining premium.

Check. At issue, $2{,}800-1{,}573.6842(1.90)=-\$190$, which matches the specified allowance with a negative sign. Rounded to the nearest cent, the net premium is $1,473.68, the Zillmer premium is $1,573.68, and the time-1 reserve is $426.32.

Worked Example: A Three-Year Policy With a Nonzero Later Reserve

A hypothetical three-year policy pays $6,100 at the end of the year of death. Annual premiums are payable in advance, $i=0$, and the conditional death probabilities by year are $q_1=0.20$, $q_2=0.20$, and $q_3=0.30$. The allowance is $100 per premium. Calculate the net premium, the issue-date allowance, and the Zillmerized reserves at times 1 and 2.

Method. Calculate the benefit present value using the probability of death in each policy year, then use the remaining conditional probabilities for each duration-specific prospective reserve.

The probabilities of death in years 1, 2, and 3 are $0.20$, $(0.80)(0.20)=0.16$, and $(0.80)(0.80)(0.30)=0.192$. The benefit present value is $6{,}100(0.20+0.16+0.192)=6{,}100(0.552)=\$3{,}367.20$.

The issue-date premium factor is $1+0.80+0.64=2.44$, so the net premium is $3{,}367.20/2.44=\$1{,}380$. The issue-date allowance is $100(2.44)=\$244$, and the Zillmer premium is $1{,}380+100=\$1{,}480$.

At time 1, the remaining expected benefits are $0.20(6{,}100)+(0.80)(0.30)(6{,}100)=1{,}220+1{,}464=\$2{,}684$. The remaining premium factor is $1+0.80=1.80$. Therefore $V_1^Z=2{,}684-1{,}480(1.80)=\$20$.

At time 2, the remaining expected benefit is $0.30(6{,}100)=\$1{,}830$, and one premium is due. Thus $V_2^Z=1{,}830-1{,}480=\$350$.

Check. The corresponding unmodified reserves are $2{,}684-1{,}380(1.80)=\$200$ and $1{,}830-1{,}380=\$450$. The Zillmerized reserves are lower by $180 and $100, respectively, matching the allowance per premium times the remaining premium factors. At issue, the modified valuation gives $3{,}367.20-1{,}480(2.44)=-\$244$. All amounts are exact to the cent.

Common Mistake: Applying the Whole Allowance at Every Duration

A frequent error is to subtract the full issue-date allowance from every later unmodified reserve. That treats acquisition costs as if the entire amount were still unamortized at each duration. Under this premium-allowance method, the reserve reduction at duration $k$ is based on the remaining allowance premiums, not automatically on the full issue-date allowance.

For example, in the three-year example, the time-1 premium factor is 1.80 and $z=\$100$. The wrong calculation subtracts the full $244 issue allowance from the unmodified reserve of $200, giving $-\$44$. The correct calculation subtracts $100(1.80)=\$180$, giving $200-180=\$20$. The issue-date allowance is $244 because there are 2.44 expected premium payments in present-value terms; it is not the time-1 adjustment.

Another timing trap is to count a premium due at time $k$ in one calculation but omit it in another. State whether the reserve is valued before or after the premium, then use that convention consistently in both the unmodified and modified valuations.

Key takeaway: Calibrate the allowance on the same premium-payment pattern used for the reserve. Add the resulting per-premium amount to the valuation premium, and calculate each Zillmerized reserve prospectively using only future premiums. The issue-date difference is the specified allowance when the unmodified basis satisfies equivalence.

Check Your Understanding

For each problem, use the stated reserve timing and round money to the nearest cent where needed.

A two-year policy has $i=0$, $q_1=q_2=0.25$, and a $4,000 death benefit paid at the end of the year of death. Premiums are annual in advance. If the issue-date allowance is $80, find the allowance per premium and the issue-date Zillmerized reserve. (Answer: $z=\$45.71$; $V_0^Z=-\$80.00$)
A two-year policy has $i=0$, $q_1=0.10$, $q_2=0.30$, and a $5,000 death benefit. Find the net premium and the unmodified reserve at time 1, immediately before the second premium. (Answer: $P=\$973.68$; $V_1=\$526.32$)
A modified basis uses an allowance of $100 per premium. At time 2, the remaining premium factor is 1.4 and the unmodified reserve is $750. Find the Zillmerized reserve. (Answer: $\$610.00$)
A contract has benefit present value $2,400 and issue-date premium factor 2.4. Find the net premium, the allowance per premium for a $120 issue-date allowance, and the Zillmerized premium. (Answer: $P=\$1{,}000.00$, $z=\$50.00$, $P^Z=\$1{,}050.00$)
At a duration with two remaining premiums, the premium annuity factor is 1.8. The unmodified reserve is $320 and the Zillmer allowance is $75 per premium. Find the modified reserve. (Answer: $\$185.00$)

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