Work out what an annuity and a perpetuity are worth today

CFA® Level I · Quantitative Methods · The Time Value of Money in Finance · QM.1.c

The idea

An annuity is a fixed payment for a fixed number of periods; a perpetuity is the same payment forever. The perpetuity is the simpler formula, PV = PMT / r, and the annuity is that perpetuity less the perpetuity you do not receive after the final payment: PV = PMT x [1 - (1 + r)^-n] / r. An annuity due pays at the start of each period rather than the end, so every payment is discounted one period less and its value is the ordinary annuity multiplied by (1 + r). Recognising a stream as an annuity turns a long summation into one calculation.

Formulas

Present value of a perpetuity

PV = PMT ÷ r

PMT: the level payment, received for ever
r: the rate per period

Level payments with no end date, first payment one period from now. A perpetuity due is worth this × (1 + r).

Present value of an ordinary annuity

PV = PMT × [1 − (1 + r)⁻ⁿ] ÷ r

PMT: the level payment each period
r: the rate per period
n: the number of payments

Payments at the END of each period. For an annuity due, multiply by (1 + r).

A practice question

From the question bank, with the reason each wrong option is wrong.

What is the present value of a perpetuity paying $60 a year when the required return is 8%?

$480.00Why not: 60 × 8 — the rate was read as the whole number 8 and multiplied in, rather than 0.08 divided by.
$750.00 — correct
$4.80Why not: 60 × 0.08. The payment was multiplied by the rate instead of divided by it, which is the perpetuity formula upside down.

PV = PMT ÷ r = 60 ÷ 0.08 = $750.00. The payments never stop, but their present values shrink fast enough that the total converges.

A harder one

What is the present value of five annual payments of $100, the first in one year, discounted at 6%?

$421.24 — correct
$500.00Why not: The five payments added up with no discounting. Waiting up to five years for them costs $78.76 of present value.
$1,666.67Why not: 100 ÷ 0.06 — the perpetuity formula, which assumes the payments never stop. These stop after five.

100 x [1 - 1.06⁻⁵] ÷ 0.06 = $421.24. The undiscounted total of $500 ignores the time value; the perpetuity figure assumes the payments never stop.

The hardest tier

How does the present value of an annuity due compare with an otherwise identical ordinary annuity?

It is lower, by a factor of (1 + r)Why not: The right factor, the wrong direction. Paying at the start of each period means waiting less, which is worth more, not less.
It is identicalWhy not: The amounts are identical; the timing is not. Every payment arrives one period sooner, so every payment is discounted one period less.
It is higher, by a factor of (1 + r) — correct

An annuity due pays at the beginning of each period, so every payment is discounted one period less. Multiplying the ordinary annuity value by (1 + r) converts between them.

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