Annuities-Due: Timing Differences and Key Formulas for Exam FM

Annuity-Due Definition

An annuity-due makes payments at the beginning of each period, one period earlier than the corresponding annuity-immediate. For n payments of 1 at the beginning of each period, the present value is a-double-dot-angle-n = (1 - v^n) / d, and the accumulated value is s-double-dot-angle-n = ((1+i)^n - 1) / d, where d = i/(1+i) is the discount rate.

The key relationship: a-double-dot-angle-n = (1+i) * a-angle-n, and similarly s-double-dot-angle-n = (1+i) * s-angle-n. Every annuity-due value is (1+i) times the corresponding annuity-immediate value because each payment is received one period earlier.

Conversion Between Due and Immediate

There are multiple equivalent expressions: a-double-dot-angle-n = 1 + a-angle-(n-1) and s-double-dot-angle-n = s-angle-(n+1) - 1. These identities are derived by considering the timing of payments. On Exam FM, you may need to use either form depending on what information is given.

For payment of R per period: PV = R * a-double-dot-angle-n and FV = R * s-double-dot-angle-n. Common applications include rent (paid at the start of each month) and insurance premiums (paid at the start of each period).

When to Use Which

On Exam FM, read each problem carefully to determine when payments occur. "Beginning of month" or "due at the start" signals annuity-due. "End of month" or "payable in arrears" signals annuity-immediate. If unsure, draw a timeline showing when each payment occurs relative to the valuation date. A common error is using the wrong annuity type, which shifts the answer by a factor of (1+i).