Zillmer Allowance Limits And Negative Reserves

Why a Zillmer Allowance Can Produce a Negative Reserve

A Zillmer adjustment recognizes part of an assumed acquisition expense through an allowance that is recovered over future premiums. Because future premium surcharges have value at the valuation date, the allowance reduces the reserve. Early in a policy, that reduction can exceed the unmodified reserve and produce a negative result. Whether that result is permitted, capped, or floored depends on the valuation rule being applied.

The important calculation is not just the reserve at issue. The amount of allowance still reflected in the reserve generally declines as fewer surcharge premiums remain. A cap on the issue allowance and a floor on the reserve are different controls: the cap reduces the allowance used in the calculation, while the floor replaces a negative reported reserve with zero.

Definition: Consider a hypothetical $n$-year contract with annual premiums due at the start of each policy year while the insured is alive, and benefits payable at the end of the year of death. Let $V_t$ be the unmodified prospective reserve at the start of duration $t$, just before any premium due at that time. Let $z$ be an additional level annual premium per policy in force, used to amortize a Zillmer allowance. Define the remaining premium annuity factor by $$ \ddot a_t^{(n)}=\sum_{k=0}^{n-t-1}v^k\,{}_kp_t, $$ where ${}_kp_t$ is the probability, conditional on being alive at duration $t$, of surviving a further $k$ years. On this basis, the Zillmerized reserve is $V_t^Z=V_t-z\ddot a_t^{(n)}$. The adjustment is the present value of the remaining surcharge premiums.

This definition uses an annual, discrete-time model with no expenses other than the allowance represented by the surcharge. The surcharge is paid only when the policy is in force at the premium date. Benefits, mortality, and interest are the same on the modified and unmodified bases. The formula does not, by itself, prescribe a regulatory allowance cap or a minimum reserve; those must be stated separately.

Deriving the Allowance Adjustment

The adjustment follows directly from the prospective-reserve definition. At duration $t$, the reserve is the expected present value of future benefits minus the expected present value of future premiums. The benefits are unchanged by the Zillmer surcharge. The only difference is that each future premium due while the policy is in force is larger by $z$.

Write the unmodified reserve.
Let $B_t$ denote the prospective value of future benefits. The unmodified reserve is $V_t=B_t-\operatorname{EPV}_t(\text{future base premiums})$.

Add the surcharge to future premiums.
The expected present value of the surcharge premiums is $z\sum_{k=0}^{n-t-1}v^k{}_kp_t=z\ddot a_t^{(n)}$.

Subtract that value from the reserve.
Since future premiums are subtracted in a prospective reserve, $V_t^Z=B_t-\operatorname{EPV}_t(\text{future base premiums})-z\ddot a_t^{(n)}=V_t-z\ddot a_t^{(n)}$.

This derivation also explains the sign. A larger surcharge means a larger value of future premiums and therefore a smaller reserve. At issue, if the unmodified reserve is zero under the equivalence principle, any positive surcharge produces a negative Zillmerized reserve unless a cap or floor changes the calculation. At later durations, the unmodified reserve may have grown enough to exceed the remaining allowance.

Negative Reserves, Caps, and Floors

A negative Zillmerized reserve is a mathematical result of the stated prospective basis; it is not automatically an error. A valuation rule may permit it, limit the allowance used to calculate it, or require a nonnegative reserve. State which rule applies before calculating. In particular, do not assume that a negative result should be changed to zero unless a floor is specified.

Allowance cap: If the issue allowance is limited to $K_{\max}$, then the effective level surcharge is $z_{\mathrm{eff}}=\min(z_{\mathrm{raw}},K_{\max}/\ddot a_0^{(n)})$. Use $z_{\mathrm{eff}}$ in the reserve adjustment at every duration. This applies the cap to the allowance at issue and keeps the assumed recovery consistent with the premiums used in the modified reserve.

Zero floor: If the rule requires a nonnegative reported reserve, calculate $V_t^Z=V_t-z\ddot a_t^{(n)}$ first, then report $V_t^{Z,\mathrm{floor}}=\max(0,V_t^Z)$. A floor applied to the reported amount does not, by itself, change the underlying premium basis or the allowance at later durations. If a rule instead requires a floor to feed into a reserve recursion, that recursion must be specified separately.

The allowance cap and the floor can be used separately or together if the valuation instructions require both. With both, first determine the effective surcharge from the cap, calculate the Zillmerized reserve using that surcharge, and then apply the floor. Applying a floor first and using the floored amount to infer a new allowance would be a different method.

Worked Calculations by Policy Duration

Worked Example: Applying a Zero Floor to an Early Negative Reserve

A hypothetical two-year term contract pays $100 at the end of the year of death. Annual premiums of $P$ are due at the start of each year while the policyholder is alive. The probability of death in year 1 is $0.10$; conditional on surviving year 1, the probability of death in year 2 is $0.20$. The effective annual interest rate is $i=0\%$, so $v=1$. Find the net premium, the Zillmerized reserves with an annual surcharge $z=\$5$, and the reserves after a stated zero floor.

Method. First calculate the net premium by equivalence, then calculate each prospective reserve and subtract the present value of remaining surcharge premiums. The common mistake is to subtract the full issue allowance at every duration rather than only the value of surcharges still due.

The probability of surviving year 1 is $0.90$. The expected present value of benefits at issue is $$ 100(0.10)+0.90(100)(0.20)=10+18=\$28. $$ The expected present value of the two possible premiums is $P(1+0.90)=1.90P$. Thus $$ P=\frac{28}{1.90}=\$14.736842\ldots. $$ At issue, equivalence gives $V_0=0$, and the remaining premium annuity factor is $\ddot a_0^{(2)}=1+0.90=1.90$. Therefore $$ V_0^Z=0-5(1.90)=-\$9.50. $$ At duration 1, the only remaining benefit has expected value $100(0.20)=\$20$, and one premium is due. Hence $V_1=20-P=\$5.263158\ldots$. Since $\ddot a_1^{(2)}=1$, $$ V_1^Z=5.263158\ldots-5(1)=\$0.263158\ldots. $$ After applying the zero floor, the reported reserves are $V_0^{Z,\mathrm{floor}}=\$0.00$ and $V_1^{Z,\mathrm{floor}}=\$0.26$, rounded to the nearest cent. The sign check is consistent: the issue allowance exceeds the zero unmodified issue reserve, while the later unmodified reserve slightly exceeds the remaining surcharge value.

Worked Example: Limiting the Issue Allowance

Use the same hypothetical contract, mortality, benefit, and interest assumptions as in the preceding example. Suppose the raw annual surcharge is $z_{\mathrm{raw}}=\$5$, but the permitted issue allowance is at most $K_{\max}=\$6$. Calculate the effective surcharge and the modified reserves at durations 0 and 1. No zero floor is imposed.

Method. Convert the issue allowance cap into a level premium surcharge using the issue annuity factor. Do not cap the surcharge at $6; the cap applies to the allowance amount, not to each premium.

The raw issue allowance is $$ K_{\mathrm{raw}}=z_{\mathrm{raw}}\ddot a_0^{(2)}=5(1.90)=\$9.50. $$ This exceeds the $6 cap. The effective surcharge is therefore $$ z_{\mathrm{eff}}=\min\left(5,\frac{6}{1.90}\right)=\$3.157894\ldots. $$ The issue reserve is $V_0=0$, so the capped modified issue reserve is $$ V_0^Z=0-3.157894\ldots(1.90)=-\$6.00. $$ At duration 1, the unmodified reserve remains $V_1=\$5.263158\ldots$, and the remaining annuity factor is 1. Thus $$ V_1^Z=5.263158\ldots-3.157894\ldots=\$2.105263\ldots. $$ Rounded to the nearest cent, the reserves are $-\$6.00$ at issue and $\$2.11$ at duration 1. The issue result equals the negative of the capped allowance, as it should when $V_0=0$. The later allowance is smaller because only one surcharge premium remains.

Worked Example: Tracking a Three-Year Reserve Path

A hypothetical three-year term contract pays $100 at the end of the year of death. Conditional probabilities of death are $0.10$ in year 1, $0.20$ in year 2, and $0.25$ in year 3. Annual premiums are due at the start of each year while the policy is in force. The effective annual interest rate is $i=0\%$. Calculate the net premium and Zillmerized reserves at durations 0, 1, and 2 for a surcharge of $z=\$4$. Also report the reserves after a zero floor.

Method. Calculate the probability of reaching each premium and benefit year, then value the remaining benefits and premiums conditionally at each duration. The main trap is using unconditional probabilities for a reserve conditional on being in force at that duration.

The probabilities of reaching the start of years 1, 2, and 3 are $1$, $0.90$, and $0.90(0.80)=0.72$. The expected present value of benefits at issue is $$ 100(0.10)+100(0.90)(0.20)+100(0.72)(0.25)=10+18+18=\$46. $$ The premium annuity factor is $1+0.90+0.72=2.62$, so the net annual premium is $$ P=\frac{46}{2.62}=\$17.557252\ldots. $$ The issue reserve is $V_0=0$, and $\ddot a_0^{(3)}=2.62$. Therefore $$ V_0^Z=0-4(2.62)=-\$10.48. $$ At duration 1, the conditional expected value of future benefits is $100(0.20)+0.80(100)(0.25)=20+20=\$40$. The remaining premium annuity factor is $1+0.80=1.80$, giving $$ V_1=40-P(1.80)=40-31.603053\ldots=\$8.396947\ldots. $$ The remaining allowance is $4(1.80)=\$7.20$, so $V_1^Z=\$1.196947\ldots$.

At duration 2, the expected value of the final-year death benefit is $100(0.25)=\$25$, and one premium remains. Thus $$ V_2=25-P=\$7.442748\ldots,\qquad V_2^Z=7.442748\ldots-4=\$3.442748\ldots. $$ Rounded to the nearest cent, the Zillmerized path is $-\$10.48,\ \$1.20,\ \$3.44$. Applying a zero floor gives $\$0.00,\ \$1.20,\ \$3.44$. The allowance falls from $10.48 at issue to $7.20 at duration 1 and $4.00 at duration 2, matching the declining number and conditional probability of surcharge premiums.

Checks and Common Traps

A reliable calculation checks the time point, the premium status, and the allowance remaining. The reserve convention here is just before a premium due at the start of the duration. Consequently, the premium due at that time is included in the prospective value. At the end of the final coverage year, there are no future premiums and no future benefits under the stated term contract, so the terminal reserve is zero.

Subtracting the issue allowance at every duration. The adjustment at duration $t$ is $z\ddot a_t^{(n)}$, not $z\ddot a_0^{(n)}$. The remaining annuity factor changes with duration.
Changing the sign. The surcharge increases future premium income, so it is subtracted from the reserve. A positive surcharge does not increase the Zillmerized reserve.
Confusing a cap with a floor. A cap changes the effective allowance and therefore the surcharge used throughout the calculation. A floor changes a negative reported reserve to zero only where the floor applies.
Mixing conditional and unconditional probabilities. A duration-$t$ prospective reserve is conditional on being in force at duration $t$. Use probabilities conditional on survival to that point.

Key takeaway: Calculate the unmodified reserve and the present value of remaining Zillmer surcharge premiums separately. Apply any stated allowance cap to the issue allowance before calculating duration-specific reserves, and apply any stated reserve floor only after obtaining the modified reserve.

Check Your Understanding

For each problem, use annual premiums due at the start of each year while the policy is in force, benefits paid at the end of the year of death, and $i=0\%$, unless stated otherwise. Reserves are measured just before a premium due.

A two-year term benefit is $200. The conditional death probabilities are $0.10$ in year 1 and $0.20$ in year 2. Find the net premium and the Zillmerized reserve at issue and duration 1 with $z=\$3$. What is the floored issue reserve? (Answer: $P=\$29.47;\ V_0^Z=-\$5.70;\ V_1^Z=\$7.53;$ floored $V_0^Z=\$0.00$.)
A three-year term benefit is $100, with conditional death probabilities $0.10,\ 0.20,\ 0.25$. Using $z=\$2.50$, find the Zillmerized reserves at durations 0, 1, and 2. (Answer: $-\$6.55,\ \$3.90,\ \$4.94$, respectively, rounded to cents.)
A two-year term benefit is $100. Conditional death probabilities are $0.20$ in year 1 and $0.10$ in year 2. Find the net premium and, with $z=\$2$, the Zillmerized reserves at issue and duration 1. Then apply a zero floor. (Answer: $P=\$15.56;\ V_0^Z=-\$3.60,\ V_1^Z=-\$7.56;$ floored reserves $=\$0.00,\ \$0.00$.)
Return to a two-year, $100 term benefit with conditional death probabilities $0.10$ and $0.20$. A raw surcharge of $5 gives an issue allowance greater than a stated cap of $5.70. Find the effective surcharge and the capped Zillmerized reserve at duration 1. (Answer: $z_{\mathrm{eff}}=\$3.00;\ V_1^Z=\$2.26$, rounded to cents.)

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