Two parties A and B may agree that A pays B a lump sum while B pays A back in annual instalments. If B needs the money and A wants the rent this is called a loan with interest. It is called an annuity when A wants a secured income for an assigned period. Annuities, as opposed to loans, were a standard way to raise public money, partly because it was possible for a government to sell security in exchange for ready cash, and partly because usury was suspect and not a proper business for the state.

Annuities are of several kinds. Perpetual annuities are straightforward loans paying annual interest to the annuitant. A terminal annuity pays an annual sum so fixed that, at the end of the designated term of n years, the capital and interest will all be repaid. A life annuity pays a set sum every year of the annuitant's life. A joint annuity on several lives pays until the death of the last survivor. The terminal annuity presents combinatorial problems: how much should one pay to receive a guaranteed £100 for ten years if the rate of interest is 6%? The life annuity adds problems in empirical probability. The fair price for £100 for life must be the same as that for a terminal annuity for n years, where n is the expectation of life. Joint annuities add a further problem in probability mathematics even if it is assumed that the duration of lives is stochastically independent. If we are more realistic and note that, far from independence, the usual joint annuity is a bet on married couples or shipmates, we require yet further statistical data on joint expectation.