By the term actual run-off triangle we shall mean the two-way tabulation—according to year of origin and year of payment—of claims paid to date, which has the following form:

where Cij is the amount paid during development year j in respect of claims whose year of origin is i.

The information relating to the area below and/or to the right of this triangle is unknown since it represents the future development of various cohorts of claims.

Now in seeking to use this triangle as a basis for projection of claims in future development years for each of the years of origin 0, 1, 2, etc., we must recognise that the entries Cij in the above triangle, being random variables, contain random deviations from their expected values uij. It is the corresponding triangle of these expected values in which we are interested, and which shall be called the expected run-off triangle.