Deriving The Zillmer Reserve Adjustment

Why the Reserve Adjustment Has a Sign

A Zillmer allowance recognizes an initial acquisition outlay by spreading an allowance for it over scheduled premiums. The reserve adjustment follows from that premium change: if the valuation premium increases, the present value of future premiums increases, so the prospective reserve decreases. The amount of the decrease depends on how many premiums remain and when they are due.

The result is easy to misapply if the premium timing is not stated. In this tutorial, reserves are first defined immediately before a premium due at the valuation time. A second convention, immediately after that premium has been received, will be handled separately.

Definition: Consider a hypothetical contract with level premiums payable annually in advance while the insured is in force, at times $0,1,\ldots,n-1$. Let $E$ be an initial acquisition allowance, and let $P$ be the level premium on the ordinary valuation basis, excluding that allowance. The Zillmer premium is $P^Z=P+\alpha$, where the positive increment $\alpha$ is chosen so that the present value of its scheduled increments at issue equals $E$. Under the same valuation basis, $V_k$ is the ordinary prospective reserve at duration $k$, and $V_k^Z$ is the reserve using $P^Z$. In the formulas below, duration-$k$ reserves are measured immediately before the premium due at time $k$.

Derivation from the Prospective Equation

Use one mortality and interest basis for both reserves. Premiums are conditional on the contract being in force when due. Write $B_k$ for the present value at time $k$ of all future benefits, conditional on being in force then, and write $\ddot a_{x+k:\overline{n-k}|}$ for the expected present value at time $k$ of one-unit annual premiums due in advance at times $k$ through $n-1$. In particular, this factor includes a premium at time $k$.

For a mortality model with survival probabilities ${}_jp_{x+k}$ and effective annual interest rate $i$, with $v=1/(1+i)$, the factor is

\ddot a_{x+k:\overline{n-k}|} = \sum_{j=0}^{n-k-1}v^j\,{}_jp_{x+k}.

The prospective reserve is the present value of future benefits less the present value of future premiums. Applying this definition to the ordinary and Zillmerized premiums gives

V_k = B_k-P\ddot a_{x+k:\overline{n-k}|}, \qquad V_k^Z = B_k-(P+\alpha)\ddot a_{x+k:\overline{n-k}|}.

Subtract the first equation from the second. The benefit present value cancels because the benefit basis is unchanged. The remaining premium increment is valued over precisely the premiums still payable from the valuation time:

V_k^Z-V_k = -\alpha\ddot a_{x+k:\overline{n-k}|}, \qquad \boxed{ V_k^Z = V_k-\alpha\ddot a_{x+k:\overline{n-k}|} }.

At issue, the allowance condition is $E=\alpha\ddot a_{x:\overline{n}|}$, so $\alpha=E/\ddot a_{x:\overline{n}|}$. Therefore the adjustment can also be written as a proportion of the initial allowance:

V_k^Z = V_k - E\frac{\ddot a_{x+k:\overline{n-k}|}}{\ddot a_{x:\overline{n}|}}.

This proportion is the part of the original allowance represented by the value of the remaining scheduled premium increments. It is not generally correct to subtract the full initial allowance at every duration.

Specify the timing.
Decide whether the reserve is measured before or after the premium due at that time. The annuity factor must match that convention.

Find the premium increment.
For a level increment recovering allowance $E$, divide $E$ by the issue value of the scheduled unit premiums.

Subtract the value of remaining increments.
The Zillmer reserve equals the ordinary reserve minus $\alpha$ times the appropriate remaining-premium annuity factor.

Check directly.
Recalculate the prospective reserve using $P+\alpha$. It must agree with the adjusted-reserve formula.

Worked Examples

Worked Example: A Two-Year Term Contract

A hypothetical two-year term insurance pays $1,000 at the end of the year of death. Annual premiums are due in advance while the insured is alive, at times 0 and 1. Use $i=10\%$, with $q_{x}=0.10$ and $q_{x+1}=0.20$. Assume no benefits other than the death benefit and an initial acquisition allowance of $E=\$200$. Find the Zillmer premium and the reserve immediately before the time-1 premium.

Method and assumptions. Use the prospective net-premium equation, with no lapses or further expenses. The one-year survival probability at issue is $0.90$, and the probability of surviving two years is $0.90(0.80)=0.72$. The discount factor is $v=1/1.10=10/11$.

The issue premium annuity factor includes the premium at time 0 and the time-1 premium if the insured survives the first year:

\ddot a_{x:\overline{2}|} = 1+0.90v = 1+\frac{9}{11} = \frac{20}{11}.

The present value of benefits at issue is the time-1 death benefit plus the time-2 death benefit, each weighted by its probability:

\begin{aligned} A &= 1000\left(0.10v+0.90(0.20)v^2\right)\\ &= 1000\left(\frac{11}{121}+\frac{18}{121}\right) = \frac{29000}{121}. \end{aligned}

Thus the ordinary level premium is

P = \frac{29000/121}{20/11} = \frac{1450}{11} = \$131.82.

The premium increment that has issue present value $200$ is

\alpha = \frac{200}{20/11} = 110, \qquad P^Z=P+\alpha = \frac{1450}{11}+110 = \frac{2660}{11} = \$241.82.

At time 1, immediately before the second premium, the remaining unit-premium annuity factor is 1. The present value of the remaining death benefit is $1000(0.20)v=2000/11$. Therefore,

V_1 = \frac{2000}{11}-\frac{1450}{11} = \$50.00, \qquad V_1^Z = V_1-\alpha = \$50.00-\$110.00 = -\$60.00.

Check. Direct valuation with the Zillmer premium gives $2000/11-2660/11=-\$60.00$, the same result. At issue, the ordinary reserve is zero by premium equivalence, while the Zillmer reserve is $0-200=-\$200.00$. The negative values are algebraic consequences of the stated prospective basis; this example does not impose a separate floor.

Common mistake avoided: adding $110$ to the ordinary reserve would give $+\$160.00$. That has the wrong sign: a larger future premium reduces, rather than increases, a prospective reserve.

Worked Example: A Certain Two-Year Benefit

A hypothetical savings contract pays $121 with certainty at time 2. Premiums are payable at times 0 and 1, with no mortality risk. The effective annual interest rate is $10\%$, and the initial acquisition allowance is $20. Find the ordinary and Zillmerized reserves at time 1 immediately before its premium.

Method and assumptions. With survival certain, every scheduled premium is payable. The issue present value of the benefit is $121v^2=121(100/121)=100$. The issue premium annuity factor is $1+v=21/11$, so the ordinary premium is

P = \frac{100}{21/11} = \frac{1100}{21} = \$52.38.

The Zillmer increment and premium are

\alpha = \frac{20}{21/11} = \frac{220}{21} = \$10.48, \qquad P^Z = \frac{1320}{21} = \$62.86.

At time 1, the benefit is one year away, so its present value is $121v=110$. The premium due at time 1 is included in the reserve convention, and the remaining unit-premium annuity factor is 1. Hence

V_1 = 110-\frac{1100}{21} = \frac{1210}{21} = \$57.62,

and

V_1^Z = V_1-\frac{220}{21} = \frac{990}{21} = \$47.14.

Check. Directly, $110-P^Z=110-1320/21=990/21=\$47.14$. Rounding to cents only at the final step gives the same reserve by either method.

Common mistake avoided: using the issue annuity factor $21/11$ for the time-1 adjustment would incorrectly subtract the value of premiums that are no longer payable. At time 1, only the one premium due then remains.

Worked Example: Before-Premium and After-Premium Reserves

A hypothetical contract pays $133.10 with certainty at time 3 and charges level annual premiums at times 0, 1, and 2. There is no mortality risk, the effective annual interest rate is $10\%$, and the chosen Zillmer increment is $\alpha=\$11$ per premium. Find the ordinary and Zillmerized reserves at time 1 immediately before and immediately after the time-1 premium. First determine the initial allowance represented by this increment.

Method and assumptions. The benefit has issue present value $133.10v^3=100$. The issue premium annuity factor is

1+v+v^2 = 1+\frac{10}{11}+\frac{100}{121} = \frac{331}{121}.

The ordinary premium and corresponding initial allowance are therefore

P = \frac{100}{331/121} = \frac{12100}{331} = \$36.56, \qquad E = 11\left(\frac{331}{121}\right) = \frac{3641}{121} = \$30.09.

Immediately before the time-1 premium, the benefit present value is $133.10v^2=110$. Two premiums remain, at times 1 and 2, so their unit annuity factor is $1+v=21/11$. Thus

V_1 = 110-\frac{12100}{331}\frac{21}{11} = \frac{13310}{331} = \$40.21,

V_1^Z = V_1-11\left(\frac{21}{11}\right) = \frac{6359}{331} = \$19.21.

Immediately after the time-1 premium is received, that premium is no longer a future cash flow. The benefit present value is still 110, but only the time-2 premium remains; its value at time 1 is $Pv$. The ordinary reserve after receipt is

V_{1+} = 110-\frac{12100}{331}\frac{10}{11} = \frac{25410}{331} = \$76.77.

The Zillmer adjustment after receipt is the value of the one remaining increment, $\alpha v=11(10/11)=10$. Consequently,

V_{1+}^Z = V_{1+}-10 = \frac{22100}{331} = \$66.77.

Check. The Zillmerized premium is $P^Z=12100/331+11=15741/331$. Directly after the time-1 premium, only its time-2 payment remains, so $110-P^Zv=110-(15741/331)(10/11)=22100/331=\$66.77$. As a further timing check, $V_{1+}^Z=V_1^Z+P^Z$: the premium is received between the before- and after-premium valuations.

Common mistake avoided: carrying the before-premium adjustment of $21 over to the after-premium valuation counts the time-1 increment again. After that premium has been received, the remaining adjustment is only $10.

Timing Conventions and Scope

If a reserve is measured immediately after a premium at time $k$, that premium is not part of future premiums. With annual premiums in advance, the remaining unit-premium value is then

\sum_{j=1}^{n-k-1}v^j\,{}_jp_{x+k},

rather than the before-premium factor that starts with $j=0$. The adjustment is minus $\alpha$ times this after-premium factor. In a model without mortality, for example, a single future premium one year later has value $v$, not 1. The corresponding reserve also changes by the premium received at the valuation time; the before- and after-premium equations must not be mixed.

The derivation assumes that both reserves use the same benefit values, interest, mortality, and premium-payment conditions, and that the only change is the uniform premium increment. If the premium increment varies by duration, its remaining present value must be calculated from those actual increments; a single $\alpha$ times an annuity factor will not generally apply. Likewise, premiums payable in arrears require an annuity factor with arrears timing.

Common Sign and Timing Errors

Reversing the sign. Since $V_k^Z=B_k-(P+\alpha)\ddot a$, the adjustment is negative for a positive increment: $V_k^Z=V_k-\alpha\ddot a$.
Subtracting the full allowance at every duration. The full $E$ is the issue value of all scheduled increments. A later reserve subtracts only the value of increments still payable.
Including a premium twice. A before-premium reserve includes the premium due at that time in its prospective equation; an after-premium reserve does not.
Changing the benefit basis accidentally. The derivation cancels benefit values only when ordinary and Zillmerized reserves use the same benefits and valuation assumptions.

The central check is to write the prospective equation under both premiums before simplifying. If the two equations do not differ only by the present value of the remaining premium increments, the timing or the definition of the allowance has changed.

Check Your Understanding

For each problem, use the stated reserve timing and subtract the value of the future Zillmer increments from the ordinary reserve.

An initial allowance of $48 is recovered through level increments over an issue premium annuity factor of 4. At a later before-premium valuation, the remaining annuity factor is 2.5. Find the increment and the reserve adjustment. (Answer: $\alpha=\$12$; adjustment $=-\$30$.)
An ordinary reserve immediately before a premium is $86. The Zillmer increment is $9, and the remaining before-premium annuity factor is 3.2. Find the Zillmerized reserve. (Answer: $57.20.)
There is no mortality risk, $i=5\%$, and immediately after a premium the only remaining increment of $20 is due one year later. Find the reserve adjustment at that time. (Answer: $-\$19.05$.)
An initial allowance of $75 is represented by an issue premium annuity factor of 5. At a later before-premium valuation, the ordinary reserve is $130 and the remaining annuity factor is 2.4. Find the Zillmerized reserve. (Answer: $94.00.)
For a certain three-year contract with premiums due at times 0, 1, and 2, $i=10\%$, and a premium increment of $11, find the adjustment immediately before time 1 and immediately after the time-1 premium. (Answer: before, $-\$21.00$; after, $-\$10.00$.)

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