From Statutory Valuation of Individual Life and Annuity Contracts, fifth edition, by Claire, Lombardi, & Summers, Volume I
In chapter 14 Universal Life, section 14.1.5 Cost of Insurance Charges, the Cost of Insurance (TC) for a Death Benefit Option 1 (Level) is given as… TC = NAR * TCM / (1-TCM)
(where all of those are properly subscripted with [x]+t)
TCM is the COI for $1 of insurance.
For DBO 2 it’s TC = NAR * TCM.
I worked with UL in a previous job & I never remember the COI for the Level DB to be calculated like that. It would be the same as for DBO 2, the only difference is how the NAR is calculated.
What do I not understand? Conceptually, why would one divide by (1-TCM)?
Maturity between ages 95 and 100 is a 7702/7702A concept.
The COI calculation works out this way if you convert from a Fackler formula to a UL accumulation formula. I believe this is shown in the LIME book. Roughly speaking, there is the affect of survivorship in DB 1 and no survivorship impact in DB 2.
The COI calculation is whatever is specified in the contract, but is usually NAR x COI_rate for DBO 1. However there have been products where the COI calc looked nothing like that. It comes down to the contract.
The formula for NAR is also contract-dependent; it will generally be of the form DB / (1+i) - AV for some (monthly) interest rate i (also specified in the contract; often based on the guaranteed minimum interests rate, but sometimes just zero).
For DBO 2 the COI formula is often either NAR x COI_rate or Face x COI_rate (which is only the same if i=0 in the NAR formula above). Ultimately it comes down to what is in the contract.
The COI rates are whatever is declared. For theoretical reasons it is common for guaranteed rates to be connected to CSO mortality by q / (1 - q), where q is a monthly CSO mortality (calculated from annual mortality in one of several ways). But current COI rates are what they are, and may only be indirectly connected to the underlying mortality assumption — only the pricing actuary would know how.
I’d also add to jraven’s post by saying on DBO 1, COI is often q/(q-1) but for DBO 2 is often simply q. When you calculate a GLP/GSP and use these assumptions for COI in an illustration, you get the same premium as you get using basic actuarial calculations, even if you use commutation functions.
On the 7702 age 100 vs age 120 front that OldTimer mentioned — once upon a time (until 2001 CSO became effective in 2007/2008 or thereabouts) the relevant statutory mortality tables only went until age 99. It was unclear under 7702 whether a policy on someone past the end of the statutory table (age 100+) would be considered by the IRS to be life insurance — the IRS often takes a very strict interpretation of what is written in the code. (GPT corridor factors only extended to age 99, and you can’t do CVAT if you don’t have mortality rates.)
That’s not a problem for newer products, now that the statutory mortality tables were extended to age 120 and some changes were made to the tax code.