Perpetuities & Deferred Annuities
A perpetuity pays forever — perpetuity-immediate has present value \(1/i\), perpetuity-due has \(1/d\). A deferred annuity whose \(n\) payments start after \(m\) periods of waiting has present value \(v^m\cdot a_n\).
By the end you'll be able to explain why an infinite stream of payments has a finite value today, compute \(1/i\) and \(1/d\) for a perpetuity, and value a deferred annuity by combining the discount factor \(v^m\) with \(a_n\).
Predict: as you drag n up, does the present value \(a_n\) keep rising forever, or does it level off? Drag n and check it against the \(1/i\) line.
Top chart: \(a_n = (1-v^n)/i\) for an annuity-immediate at \(i = 5\%\), climbing toward the perpetuity line \(1/i\) as \(v^n\to 0\). Bottom chart: the same \(n\) payments, but now delayed by m periods — every payment bar shifts right, and the present value bar at today (t = 0) shrinks by an extra factor of \(v^m\).
a₂₀ = 12.4622 → 1/i = 20.0000 as n→∞
1/i (immediate) = 20.0000 · 1/d (due) = 21.0000 · v⁵·a₂₀ = 9.7645
Push the number of payments \(n\) in an annuity-immediate to infinity and something surprising happens: the present value doesn't blow up — it settles down. That's a perpetuity. Push the whole stream later in time instead, and you get a deferred annuity: the same shape, just discounted by one more factor.
Take the annuity-immediate formula \(a_n = (1-v^n)/i\) and let \(n\to\infty\). Since \(0 < v < 1\), \(v^n \to 0\), so \(a_\infty = 1/i\) — the perpetuity-immediate value. The due version follows the same limit: \(\ddot{a}_\infty = 1/d\). Because \(v + d = 1\) gives \(1/d = 1/i + 1\), the due perpetuity is worth exactly one extra payment more than the immediate one — that extra payment is the one made today, at time 0, that the immediate version doesn't have. There is no accumulated-value counterpart: an infinite stream never has a finite FUTURE value, only a finite present value. For a deferred annuity, an \(n\)-payment annuity-immediate whose first payment falls at time \(m+1\) is worth \(a_n\) as of time \(m\); discount that back \(m\) more periods to get \(PV = v^m\cdot a_n\), often written \({}_m|a_n\).
An endowment or a share of preferred stock that pays a level amount forever is valued exactly like a perpetuity — deposit \(1/i\) today, and each period's interest covers the payment while leaving the principal untouched forever. A pension that won't start paying out until an employee retires \(m\) years from now is a deferred annuity: value the payments as an ordinary annuity as of the retirement date, then discount that lump sum back to today by \(v^m\).
An endowment pays 20000 per year forever, first payment in one year, \(i = 4\%\). Its value today is the perpetuity-immediate limit: \(PV = 20000/0.04 = \) 500000.
A deferred annuity: payments of 100 per year for 5 years begin at the end of year 4 (deferred 3 years), \(i = 6\%\). First find the building blocks: \(v^3 = 0.839619\) and \(a_5 = (1-v^5)/0.06 = 4.21237\). Now finish it: \(PV = 100 \times 4.21237 \times 0.839619 = \) ____
Reveal the answer
\(100 \times 4.21237 \times 0.839619 \approx\) 353.68. Notice the deferral exponent is 3, not 4 — \(a_5\) already values the 5 payments as of time 3, one period before the first payment at time 4, so only 3 more periods of discounting are needed to reach today.
More info — why the deferral exponent is m, not m+1
A clean way to check the deferral exponent: the deferred stream plus a "front" annuity covering times \(1,\ldots,m\) together make up the full \((m+n)\)-term annuity, so \(a_{m+n} = a_m + v^m\cdot a_n\). Rearranging, \(v^m\cdot a_n = a_{m+n} - a_m\) — the deferred value is just the full annuity minus the part you'd already have received before the deferral ended. Since \(a_n\) is measured one period before its own first payment, discounting it back to today only takes \(m\) steps, not \(m+1\). See the OpenStax perpetuity section in Dive deeper below for the finite-value argument behind \(1/i\).
Check your understanding
A trust pays 3000 per year forever, first payment in one year, \(i = 8\%\). What is its value today?
As the number of payments \(n\) in an annuity-immediate grows without bound, what happens to its present value \(a_n\)?
Payments of 200 at the end of each year for 6 years begin at the end of year 5 (deferred 4 years), \(i = 5\%\). Find the present value today.
Which correctly relates the perpetuity-due value \(\ddot{a}_\infty\) to the perpetuity-immediate value \(a_\infty\)?
Recap
- A perpetuity-immediate is the limit of \(a_n=(1-v^n)/i\) as \(n\to\infty\): since \(v^n\to 0\), its present value is \(1/i\).
- A perpetuity-due has present value \(1/d = 1/i + 1\) — one extra (today's) payment more than the immediate version.
- A deferred annuity whose \(n\) payments start after \(m\) periods of waiting has present value \(PV = v^m\cdot a_n\).
- An infinite payment stream has no finite accumulated (future) value — only a finite present value, and only when \(i>0\).
Dive deeper
- OpenStax Principles of Finance — 8.1 Perpetuities Motivate the perpetuity PV=C/r formula and why an infinite stream has finite value.
- Marcel Finan — Exam FM study guide (PDF) Derive perpetuity limits and deferred-annuity valuation v^m·aₙ.
Sources
- Perpetuities and Deferred Annuities