Discussions of papers already published: "Calculating funding premiums for universal life insurance," Calvin Cherry, April 2000

North American Actuarial Journal, Jul 2001 by Lange, David R, Simkins, Betty J

In an overview of common formulas and iterative approaches used to calculate universal life (UL) premiums given a desired funding goal, Cherry notes several formula method limitations and the importance of iterative approach solution convergence. Our discussion provides an Excel spreadsheet application) that is relatively flexible and overcomes some of the formula limitations identified by Cherry.

Life insurance premium determination, funding, is often included in finance curricula as part of a risk management and insurance or personal finance course. Class time constraints, however, generally do not allow for the development of the sophisticated mathematical and statistical models common to actuarial science as highlighted by Cherry. Although the recursive nature of the formula solution may not be fully incorporated and, thus, appreciated by students, there are pedagogic benefits of modeling life insurance funding in a spreadsheet application in the above courses.

Original Data Input

The Excel spreadsheet model presented in Table 1 is quite flexible. Once the original data is entered, premiums based on changes in gender, age, payment mode, number of payments, expenses, fees, guaranteed or current mortality charges, guaranteed, current and even yearly interest rates, policy type, and base death benefits may be estimated with minimum difficulty. Cells A1 to O4 contain the original UL funding data input area and cells A6 to 08 contain associated codes. For pedagogic purposes the input data is designed to be self-explanatory. For clarity, we provide a brief overview of specific input cell references used in Table 1.

* Month Adjustment (134) changes the AGE (column C) to reflect the birth month. This may be used to change the mortality charge from the age last birthday (134 = 0) to age nearest birthday.

* Insured age at policy issue, (C2).

* #Years (C4) is the number of years premiums are payable.

* Premium (D2) is arbitrarily set at $1,000.

* Payment MODE (D4) includes the payment frequency code for monthly to yearly (C7:D8).

* Male/female (E2), codes (E7:ES).

* Guar/Curr COI (E4), codes (E7:E8) selects either guaranteed or current mortality charges to be used in the application. The original data input is set to current mortality charges.

* Guar. Rate (F2) is the guaranteed interest rate. Monthly guaranteed interest rate factor is (F4). * Expense (G2) and Fee (112) contain the policy

expense and fee as a percentage of premium and number of years (#Years) each is applied, (G4) and (H4).

* Curr./Yearly (I4), codes (I7:I8), selects either a constant current interest rate or changing yearly rates. Yearly Rates are provided in column Q. The original data input is set to the current rate.

* Curr. Rate (J2) is the current interest rate. Monthly current interest rate factor is (J4).

* Age (K2:K3) and Accumulation Values (L2:L3) are used in the premium calculation process discussed below.

* Death Benefit, (02).

* Policy Type (04), codes (07:08), selects a Type A or Type B policy death benefit.

Formulas

The spreadsheet formulas are based on a retrospective life insurance valuation model as identified in Jordan (1991), Bowers et al. (1997), and Black and Skipper (1994). The spreadsheet is primarily intended for pedagogic purposes and, thus, the formula sequence is intentionally suggested by the column headings. As is standard practice, the spreadsheet calculates monthly accumulation values, even though premiums may be paid annually, semi-annually, quarterly, or monthly. The policy cash accumulation value (column L) is equal to the beginning period accumulation value (column L, ending period for prior month), plus premium payments received (column D), less the cost of insurance (column F), less expenses (column G) and fees (column H), plus interest (column K). The cost of insurance (column F) is equal to the mortality charge per thousand times the amount of insurance purchased divided by 1,000. The amount of insurance purchased for Type A is the present value of the base death benefit or estimated Section 7702 death benefit, less the previous period ending accumulation value, plus premiums received, minus expenses and fees. The amount of insurance purchased for type B is the present value of the base death benefit.

Valuation Assumptions

The following policy valuation assumptions are only used as a pedagogic example and do not suggest a standard policy design. Actual UL policy interest rates, mortality charges, expenses and fees vary across policies (see Black and Skipper 1994 and Gold 1994 or refer to the UL section in A.M. Best's Flitcraft Compend). Chung and Skipper (1987) found a considerable dispersion of accumulation values by stated interest rates and between stated interest rates. In one study of 170 UL policies, the current interest rate varied from 5% to 8.796, expenses and fees ranged from 2% to 14% of premium, and the mortality charge as a percentage of the 1980 CSO mortality table was 209-95% (see Lange, Lucy-Bouler, and Johnson 1995).