Annuities & Perpetuities

Valuing level streams of cash flows

Finance FundamentalsThe Time Value of MoneyFree preview

⏱️ About 16 min

A pension pays you $500 every year for 20 years. Instead of discounting twenty cash flows one by one, is there a shortcut?

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The big idea: A level stream of equal payments has a closed-form present value; letting the stream run forever collapses it to the simple perpetuity formula.

🎯 By the end, you'll be able to

Apply the ordinary-annuity present-value formula.
Value a perpetuity as the limiting case of an annuity.
Recognize where annuities and perpetuities show up in practice.

📎 Helpful to know first

The lessons on future and present value.

A stream of equal payments

An ordinary annuity pays a fixed amount $PMT$ at the end of each period for $n$ periods. Rather than discount each payment separately and add them, the geometric series collapses to a closed form:

\[ PV = PMT \times \dfrac{1 - (1 + r)^{-n}}{r} \]

Letting it run forever

A perpetuity pays $PMT$ every period forever. As $n \to \infty$ the term $(1+r)^{-n} \to 0$, and the annuity formula simplifies to a strikingly simple result:

\[ PV_{\text{perpetuity}} = \dfrac{PMT}{r} \]

✅ Where you see these

Mortgages, car loans, and bond coupons are annuities; a fixed-dividend share or an endowment payout behaves like a perpetuity. The same two formulas cover a huge range of instruments.

🎮 Annuity Value Explorer LIVE

Predict first: as you increase the number of payments, the present value rises but flattens out. What value does it flatten toward?

Present value of a level annuity as payments accumulate, with the perpetuity level marked.

📝 Worked example: A pension pays $500 at the end of each year for 20 years. At a discount rate of 5%, what is its present value?

Inputs: $PMT = 500$, $r = 0.05$, $n = 20$.
Compute $(1.05)^{-20} = 0.376889$, so $1 - 0.376889 = 0.623111$.
Divide by $r$: $0.623111 / 0.05 = 12.46223$ (the annuity factor).
Multiply: $PV = 500 \times 12.46223 = 6231.11$.

✓ PV = $6,231.11

✏️ Practice: What is the present value of $1,200 received at the end of each year for 10 years at a 6% discount rate?

💡 Hint

Annuity factor $= (1 - 1.06^{-10}) / 0.06 = 7.360087$; multiply by 1200.

Answer

PV = $8,832.10

Check your understanding

1. As the number of annuity payments $n$ grows without bound, the present value approaches:

The term $(1+r)^{-n}$ vanishes, leaving $PV = PMT/r$, the perpetuity formula.

2. A perpetuity pays $500 per year and the discount rate is 5%. Its present value is:

$PV = PMT/r = 500 / 0.05 = 10{,}000$.

✅ Key takeaways

An ordinary annuity has present value $PMT \times (1 - (1+r)^{-n})/r$.
A perpetuity is the $n \to \infty$ limit, worth simply $PMT/r$.
These two formulas value loans, coupons, pensions, and endowments.

➡️ So far $r$ has been a clean per-period rate. But quoted rates hide how often interest compounds - the difference between nominal and effective rates is next.

Want to test yourself on this? Try the Finance Fundamentals Aptitude test →

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Educational content only. This course explains finance concepts and the math behind them; it is not financial, investment, tax, accounting, or trading advice, makes no recommendations, and predicts no returns, prices, or outcomes. All figures are illustrative teaching examples. Consult a qualified professional before making financial decisions.