Special Applications — Annuity Due, Deferred Annuity and Perpetuity

Beyond the plain ordinary annuity, three special cases occur often enough in business to be treated on their own. Each is obtained by a small, logical adjustment to the two master formulae of §2 and §3 — there is nothing new to memorise if the reasoning is understood.

1. Annuity due (payments in advance). When each payment is made at the beginning of its period, every payment is invested (or discounted) one period differently from the ordinary case. Multiply the ordinary result by $(1+i)$:

Rent and insurance premiums paid in advance are the everyday examples.

2. Deferred annuity. When the payments do not start at the end of the first period but only after $m$ idle periods of deferment, the ordinary present-value factor is discounted back over those extra $m$ periods:

A loan on which no repayment is due for the first few years (a repayment holiday) is a deferred annuity.

3. Perpetuity — an annuity that never ends. If an annuity continues forever, the number of payments $n \\to \\infty$. In the present-value factor, $(1+i)^{-n} \\to 0$ as $n$ grows without limit, so the entire bracket collapses to $\\dfrac{1}{i}$, giving the strikingly simple result:

$V_{\\text{perpetuity}} = \\dfrac{P}{i}$ …