Which type of annuity cannot be funded under any circumstances?

The type of annuity that cannot be funded under any circumstances is the periodic premium immediate annuity. This type of annuity requires periodic premium payments, but it also begins to make payouts immediately after the initial premium payment. Since immediate annuities start making payments right away, there is no time to accumulate those periodic premiums; therefore, it doesn't make sense to fund them on a periodic basis.

In contrast, a single premium immediate annuity is funded with just one lump sum payment and begins payments immediately, which is a feasible arrangement. Similarly, both the periodic premium deferred annuity and the single premium deferred annuity allow for accumulation of funds before the annuity payments begin, making them viable options for funding. The immediate nature of payments in a periodic premium immediate annuity disrupts the capacity to fund it correctly compared to the other types, leading to its classification as the unviable option when considering funding structures.