What Is the Difference Between Annuity and Perpetuity?
Understanding the difference between annuity and perpetuity is essential for anyone studying finance, preparing for exams, or making long‑term investment decisions. Both concepts describe streams of cash flows, but they differ in duration, valuation formulas, and practical applications. This article breaks down each term, highlights their key distinctions, and shows how they are used in real‑world scenarios.
Definition of Annuity
An annuity is a series of equal payments made at regular intervals over a finite period. Worth adding: the payments can occur at the end of each period (ordinary annuity) or at the beginning (annuity due). Common examples include mortgage repayments, car loans, retirement pensions, and insurance premiums.
Key characteristics of an annuity
- Finite horizon – the cash flow stops after a predetermined number of periods (n).
- Equal payment amount – each installment is the same (PMT).
- Regular interval – payments happen monthly, quarterly, annually, etc.
- Present value depends on n – the longer the term, the higher the present value, but it converges as n grows.
Present value formula (ordinary annuity)
[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} ]
where
- (PMT) = payment per period
- (r) = interest rate per period
- (n) = total number of periods
Future value formula (ordinary annuity)
[ FV = PMT \times \frac{(1 + r)^{n} - 1}{r} ]
Definition of Perpetuity
A perpetuity is a stream of equal cash flows that continues forever—that is, it has an infinite number of periods. Because the payments never end, the present value of a perpetuity is simpler to calculate than that of an annuity. Typical textbook examples include preferred stock dividends (if assumed constant forever) and certain government bonds known as consols.
Key characteristics of a perpetuity
- Infinite horizon – payments continue indefinitely.
- Equal payment amount – each cash flow is identical (C).
- Regular interval – same frequency as annuities (usually annual).
- Present value independent of time – the value depends only on the payment size and discount rate.
Present value formula (perpetuity)
[ PV = \frac{C}{r} ]
where
- (C) = cash flow per period
- (r) = discount rate per period
Note: A growing perpetuity, where cash flows increase at a constant rate (g), uses the formula (PV = \frac{C}{r - g}) (provided (r > g)).
Key Differences Between Annuity and Perpetuity
| Aspect | Annuity | Perpetuity |
|---|---|---|
| Duration | Finite (n periods) | Infinite |
| Present value formula | (\displaystyle PMT \times \frac{1 - (1 + r)^{-n}}{r}) | (\displaystyle \frac{C}{r}) |
| Future value | Exists (compounded over n) | Not defined (no endpoint) |
| Sensitivity to n | PV rises with n, asymptotically approaches perpetuity value | No n term; PV fixed |
| Typical use cases | Loans, mortgages, retirement payouts, lease payments | Preferred stock valuation, consols, theoretical models |
| Risk considerations | Subject to reinvestment risk over finite horizon | Assumes constant cash flow forever; more sensitive to discount rate changes |
The most intuitive way to see the relationship is to let the number of periods (n) in an annuity grow very large. As (n \to \infty), the annuity present value formula converges to the perpetuity formula:
[ \lim_{n\to\infty} PMT \times \frac{1 - (1 + r)^{-n}}{r} = \frac{PMT}{r} ]
Thus, a perpetuity can be viewed as an annuity with an endless horizon And it works..
Types of Annuities and Perpetuities
Annuity Variants
- Ordinary Annuity – payments at period end.
- Annuity Due – payments at period beginning; PV and FV are multiplied by ((1+r)).
- Deferred Annuity – payments start after a delay period.
- Variable Annuity – payment amount fluctuates based on underlying investments.
- Fixed Annuity – payment amount is constant.
Perpetuity Variants
- Level Perpetuity – constant cash flow (the basic formula above).
- Growing Perpetuity – cash flow grows at a constant rate (g); PV = (C/(r-g)).
- Delayed Perpetuity – first payment occurs after a deferment period; PV is discounted back accordingly.
Practical Examples
Example 1: Calculating the Present Value of a Mortgage (Annuity)
Suppose you take out a 30‑year fixed‑rate mortgage with monthly payments of $1,200 and an annual interest rate of 4% (0.3333% per month) Simple, but easy to overlook..
- (PMT = 1{,}200)
- (r = 0.04/12 = 0.003333)
- (n = 30 \times 12 = 360)
[ PV = 1{,}200 \times \frac{1 - (1 + 0.003333)^{-360}}{0.003333} \approx 1{,}200 \times 209 Worth keeping that in mind..
The loan amount you can afford today is roughly $251k.
Example 2: Valuing a Preferred Stock (Perpetuity)
A company issues preferred stock that pays a quarterly dividend of $2.50 indefinitely. Investors require a 6% annual return (1.5% per quarter).
- (C = 2.50)
- (r = 0.06/4 = 0.015)
[ PV = \frac{2.50}{0.015} \approx $166.67 ]
Each share is worth about $166.67 based on the perpetuity assumption But it adds up..
Example 3: Comparing a 20‑Year Annuity to a Perpetuity
Assume an annual payment of $5,000 and a discount rate of 5%.
- Annuity (20 years)
[ PV_{20} = 5{,}000 \times \frac{1 - (1.Because of that, 05)^{-20}}{0. 05} \approx 5{,}000 \times 12.
- Perpetuity
[ PV_{\infty} = \frac{5{,}000}{0.05} = $100
