An annuity in arrears is another name for an ordinary annuity, a series of equal payments made or received at the end of each period for a fixed number of periods. Most loan and mortgage repayments and bond interest coupons follow this pattern. The main entry gives the present value and future value formulas with worked examples.
What "in arrears" means here
In everyday use, being in arrears means being behind on payments. In annuity mathematics the phrase has no suggestion of default: it simply means that each payment falls due at the end of the period it relates to, after the period has run. The term is common in actuarial science, insurance and British textbooks, while American finance texts usually say ordinary annuity.
The opposite timing is an annuity in advance, better known as an annuity due, where each payment comes at the start of the period. Because payments in arrears arrive one period later, an annuity in arrears is worth slightly less in present value terms than an otherwise identical annuity in advance.
Standard tables and spreadsheet functions for the present value of an annuity usually assume payments in arrears unless told otherwise. A mortgage with monthly payments starting one month after the loan is made is a typical case, and it is valued using the ordinary annuity formula with a monthly rate and the number of monthly payments.