Annuities and Perpetuities

Core Idea

An annuity is a series of equal cash flows paid at regular intervals; its present value is PV = C × [1 − (1+r)^−t] / r. A perpetuity pays equal cash flows forever and has the elegantly simple formula PV = C/r, derived by taking the annuity formula to the limit as t → ∞. A growing perpetuity, where payments grow at constant rate g, gives PV = C/(r−g), the foundation of the Gordon Growth Model for equity valuation. These formulas are derived by summing geometric series and appear throughout finance in pricing bonds, mortgages, preferred stock, and endowments.

How It's Best Learned

Derive the perpetuity formula as the limit of the annuity formula to see where C/r comes from. Apply annuity formulas to compute monthly mortgage payments and retirement income streams. Recognize the growing perpetuity as a direct precursor to dividend discount stock valuation.

Common Misconceptions

• Perpetuities seem impractical but government consols and preferred stocks closely approximate them in real markets.
• The growing perpetuity formula breaks down entirely when g ≥ r, producing negative or infinite values — students must check this condition before applying.

Explainer

You already know how to discount a single future cash flow back to present value: divide by (1+r)^t. An annuity and a perpetuity are simply organized collections of such cash flows — the intellectual challenge is finding a compact formula instead of summing thousands of individual discounting calculations.

Start with a perpetuity: a payment of $C every period, forever. In year 1 you receive C/(1+r), in year 2 you receive C/(1+r)², in year 3 you receive C/(1+r)³, and so on forever. This is a geometric series with first term C/(1+r) and ratio 1/(1+r). From your geometric series prerequisite, you know the sum of an infinite geometric series a + ar + ar² + … = a/(1−r) whenever |r| < 1. Applying this formula gives PV = [C/(1+r)] / [1 − 1/(1+r)] = C/r. The elegance is striking: an infinite stream of payments collapses to a single fraction. A UK government consol paying £50 per year when the discount rate is 5% is worth exactly £1,000 — nothing more to compute.

An annuity is a perpetuity that stops after T periods. You can obtain its formula by thinking of the annuity as a perpetuity starting today minus a perpetuity starting at time T (whose value today is discounted back T periods): PV = C/r − [C/r] × 1/(1+r)^T = C × [1 − (1+r)^−T] / r. The bracketed term is the annuity factor — a number between 0 and 1 that scales the perpetuity value down for finite lives. Mortgage calculations are annuity problems in disguise: you borrow a lump sum PV today and repay equal monthly amounts C over 30 years; solving for C given PV, r, and T gives the monthly payment formula used by every bank.

The growing perpetuity extends the model to payments that grow at rate g each period: C in period 1, C(1+g) in period 2, C(1+g)² in period 3. Discounting each and summing the resulting geometric series (now with ratio (1+g)/(1+r)) yields PV = C/(r−g), valid only when r > g. This formula is the foundation of the Gordon Growth Model for stock valuation, which you will encounter next. Intuitively, faster growth means more valuable future payments, so the denominator shrinks and the value rises. But when g approaches r, the denominator approaches zero and value approaches infinity — growth cannot exceed the discount rate forever in a finite economy, so this boundary is economically meaningful, not just a mathematical quirk.