Annuities & Perpetuities
Valuing level streams of cash flows
⏱️ 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.
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: 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?
- 1. Inputs: $PMT = 500$, $r = 0.05$, $n = 20$.
- 2. Compute $(1.05)^{-20} = 0.376889$, so $1 - 0.376889 = 0.623111$.
- 3. Divide by $r$: $0.623111 / 0.05 = 12.46223$ (the annuity factor).
- 4. 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.
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