Annuities

You are given a series of payments — level, arithmetically increasing, or geometrically growing — and a valuation date.

Identify whether the annuity is due or immediate, then write the basic level-annuity PV, a-angle-n or ä-angle-n. For arithmetic increases, use (Ia)-angle-n or its decreasing counterpart; for geometric growth, divide payments by the growth-adjusted rate (1 + i) / (1 + g) - 1 and value as a level annuity. Always re-anchor the cash flow timeline to the valuation date.

a-angle-n = (1 - v^n) / i; ä-angle-n = a-angle-n · (1 + i).

Payments occur m times per year but interest is quoted at an annual effective rate (or vice versa).

Either convert the rate to a per-period effective rate matching the payment frequency, or use the m-thly annuity formulas a^{(m)}-angle-n in terms of i and i^{(m)}. The two approaches give identical answers — pick whichever the calculator handles cleanly. Double-check whether the annuity is "due" (start of period) or "immediate" (end of period).

a^{(m)}-angle-n = (1 - v^n) / i^{(m)}.

A perpetuity or a deferred annuity is mixed with another stream and you must price the combination as of a given date.

Decompose the cash flows into pieces you can value with closed forms: perpetuity-immediate is 1/i, perpetuity-due is 1/d, and a deferred annuity is v^k times the standard annuity. Add the present values; do not try to handle the combined stream as a single formula.

Perpetuity-immediate PV = 1/i; Perpetuity-due PV = 1/d = (1 + i)/i.