Life Actuarial Science & Statutory Accounting
Zillmerized Policy Reserve & Acquisition DAC Formula
Calculate Net Level Premium Life Reserve V_t = SA * (1 - a_{x+t} / a_x) and August Zillmer’s (1863) acquisition-adjusted reserve V_t^Z = V_t - z_dac * SA * (a_{x+t} / a_x) with zero-floor statutory flooring.
Resolve Year-1 life insurance statutory surplus strain via Zillmerization!
Using Euler’s classic life-annuity identity A_x = 1 - d * a_x (where d = i / (1 + i) is the rate of discount), a whole-life or endowment policy of Sum_Assured SA with initial annuity-due factor a_x (at issue) and remaining annuity-due factor a_{x+t} < a_x (at duration t) carries exact Prospective Net Level Premium Reserve V_t = SA * (1 - a_{x+t} / a_x)! However, because insurers pay heavy upfront agent commissions z_dac * SA in Year 1, booking full unadjusted V_1 creates severe statutory "new-business surplus strain"! German actuary August Zillmer (1863) solved this by loading acquisition amortization z_dac * SA / a_x into the renewal net premium, reducing policy reserve by Unamortized DAC = z_dac * SA * (a_{x+t} / a_x) to yield Zillmer Reserve V_t^Z = SA * (1 - (1 + z_dac) * (a_{x+t} / a_x)) (floored at 0 under statutory non-negative reserve rules)!
Verify on Sum_Assured SA = $100,000 (B2), Issue_Annuity_Due a_x = 16.00 (B3), Attained_Duration_Annuity a_{x+t} = 12.00 (B4 -> a_{x+t} / a_x = 0.7500), and Zillmer_Quota z_dac = 0.0400 (4.00% of sum assured = $4,000 upfront acquisition cost, B5): Net Level Premium Reserve V_t is 100,000 * (1 - 12 / 16) = $25,000.00; Unamortized Acquisition DAC is 0.04 * 100,000 * (12 / 16) = $3,000.00; Exact Mathematical Zillmer Reserve is 25,000.00 - 3,000.00 = $22,000.00; and Statutory Floored Reserve MAX(0, V_t^Z) = $22,000.00! Test early Duration 1 where a_{x+1} = 15.68 (1 - 15.68/16 = 2.00% < z_dac): mathematical V_1^Z turns negative (-$1,920.00), triggering Statutory Zero Floor = $0.00!
| Actuarial Reserve Metric (SA=$100k, a_x=16, a_{x+t}=12, z=4%) | Formula | Verified Output |
|---|---|---|
| Net Level Premium Reserve (V_t) | SA * (1 - a_{x+t} / a_x) | $25,000.00 |
| Unamortized Zillmer DAC Deduction | z_dac * SA * (a_{x+t} / a_x) | $3,000.00 |
| Unfloored Zillmer Reserve (V_t^Z) | V_t - Unamortized_DAC | $22,000.00 |
| Statutory Zero-Floored Reserve | MAX(0, V_t^Z) | $22,000.00 ($0 at a_{x+1}=15.68) |
Net Level Reserve vs Zillmer Reserve Formula
Notice that at maturity or advanced age as a_{x+t} -> 0, Unamortized DAC -> 0 and V_t^Z converges 100% to full Sum Assured SA!
=LET(sa_vl, B2, ax_in, B3, ax_at, B4, z_dc, B5, IF(OR(sa_vl <= 0, ax_in <= 1, ax_at < 0, ax_at > ax_in, z_dc < 0, z_dc > 0.1), "Invalid Zillmer Input", ROUND(sa_vl * (1 - (1 + z_dc) * (ax_at / ax_in)), 2)))Statutory Zero-Floored Reserve Formula
Prudential solvency regimes (Solvency II / statutory valuation) prohibit booking negative policy liabilities as an asset without surrender-value recovery — wrap V_t^Z in MAX(0, ...)!
=IF(ISTEXT(E4), E4, MAX(0, E4))Try the example
Use the original values or change them to test your understanding.