Increasing and Decreasing Annuities for Exam FM

Increasing Annuity-Immediate

An increasing annuity-immediate (Ia)-angle-n pays 1 at time 1, 2 at time 2, ..., n at time n. Its present value is (Ia)-angle-n = (a-double-dot-angle-n - n*v^n) / i. An alternative formula is (Ia)-angle-n = (a-angle-n - n*v^n) / i + a-angle-n, but the first form is more commonly used on Exam FM.

For payments that increase by a constant amount Q (not necessarily 1), the present value is P * a-angle-n + Q * (Ia)-angle-n - Q * a-angle-n, where P is the first payment. Alternatively, use the general formula for an arithmetic annuity: PV = (P + Q/i) * a-angle-n - Q*n*v^n / i.

Decreasing Annuity-Immediate

A decreasing annuity-immediate (Da)-angle-n pays n at time 1, n-1 at time 2, ..., 1 at time n. Its present value is (Da)-angle-n = (n - a-angle-n) / i. A useful identity: (Ia)-angle-n + (Da)-angle-n = (n+1) * a-angle-n. This identity can save time on Exam FM if one of the two values is easier to compute.

Geometric Annuities

When payments grow geometrically (each payment is (1+g) times the previous), the present value of n payments starting at 1 is: PV = (1 - ((1+g)/(1+i))^n) / (i - g) when i is not equal to g, and PV = n/(1+i) when i = g. This formula handles salary-linked payments and inflation-adjusted annuities, both common on Exam FM. Always verify whether the growth rate g applies to each payment relative to the previous one and whether the first payment occurs at time 0 (due) or time 1 (immediate).