Annuities and Perpetuities Explained

Monthly rent, a pension cheque, a savings plan, a car instalment: many of the cash flows in our lives are a fixed amount paid at regular intervals. In finance such a stream is called an annuity, and when it continues forever it becomes a perpetuity. Once you master these two ideas, you can value pensions, plan savings, price loans and value shares with a few short formulas instead of long tables.

What Is an Annuity?

An annuity is a series of equal payments made at equal time intervals for a fixed number of periods. Three features define it:

• Payment size \( PMT \), the same every period.
• Rate per period \( r \), matching the payment frequency.
• Number of payments \( n \).

There are two main types, depending on when payments fall:

• Ordinary annuity (annuity in arrears): payments come at the end of each period. Loan EMIs and most bond coupons work this way.
• Annuity due (annuity in advance): payments come at the start of each period. Rent and insurance premiums are typical examples.

Present Value of an Ordinary Annuity

The present value is the sum of each payment discounted back to today:

$$ PV = \frac{PMT}{1+r} + \frac{PMT}{(1+r)^2} + \cdots + \frac{PMT}{(1+r)^n} $$

This is a geometric series with first term \( PMT/(1+r) \) and common ratio \( 1/(1+r) \). Using the geometric sum formula and simplifying:

$$ PV = PMT \cdot \frac{1-(1+r)^{-n}}{r} $$

The fraction is called the present value annuity factor, often written \( a_{\overline{n}|r} \).

Future Value of an Ordinary Annuity

Each payment is compounded forward to the end of period \( n \). The first payment grows for \( n-1 \) periods and the last earns nothing:

$$ FV = PMT\left[(1+r)^{n-1} + (1+r)^{n-2} + \cdots + 1\right] = PMT \cdot \frac{(1+r)^n - 1}{r} $$

Converting to an Annuity Due

In an annuity due, every payment arrives one period earlier, so each is worth one extra period of interest. Simply multiply by \( (1+r) \):

$$ PV_{\text{due}} = PV_{\text{ordinary}} \times (1+r), \qquad FV_{\text{due}} = FV_{\text{ordinary}} \times (1+r) $$

Worked Example 1: Comparing Ordinary and Due

Sana saves PKR 10,000 every year for 5 years at 10% per year. Find the present and future values if she pays (a) at the end of each year and (b) at the start of each year.

Step 1: Annuity factors. With \( r = 0.10 \) and \( n = 5 \), \( (1.1)^5 = 1.61051 \).

$$ \text{PV factor} = \frac{1 - 1/1.61051}{0.10} = \frac{1 - 0.620921}{0.10} = 3.790787 $$

$$ \text{FV factor} = \frac{1.61051 - 1}{0.10} = 6.1051 $$

Step 2: Ordinary annuity.

$$ PV = 10{,}000 \times 3.790787 = \text{PKR } 37{,}907.87 $$

$$ FV = 10{,}000 \times 6.1051 = \text{PKR } 61{,}051.00 $$

Step 3: Annuity due. Multiply by 1.10.

$$ PV_{\text{due}} = 37{,}907.87 \times 1.10 = \text{PKR } 41{,}698.65 $$

$$ FV_{\text{due}} = 61{,}051.00 \times 1.10 = \text{PKR } 67{,}156.10 $$

Check: \( FV = PV(1+r)^n \) should hold. Indeed \( 37{,}907.87 \times 1.61051 = 61{,}051.00 \). Paying a year earlier adds exactly 10% to both values.

Worked Example 2: Retirement Savings Plan

Imran, aged 35, invests PKR 20,000 at the end of every month in a fund earning 12% per year compounded monthly. How much will he have at 60?

The monthly rate is \( 0.12/12 = 0.01 \) and the number of payments is \( 25 \times 12 = 300 \).

$$ FV = 20{,}000 \times \frac{(1.01)^{300} - 1}{0.01} $$

Compute \( (1.01)^{300} = 19.788466 \). Then

$$ FV = 20{,}000 \times \frac{18.788466}{0.01} = 20{,}000 \times 1{,}878.8466 = \text{PKR } 37{,}576{,}932.52 $$

Imran contributes only \( 20{,}000 \times 300 = \text{PKR } 6{,}000{,}000 \) himself. More than PKR 31.5 million is compound growth, which shows why starting early matters so much.

Worked Example 3: Valuing a Pension

A retiring officer is offered either a lump sum or a pension of PKR 50,000 per month for 20 years. If money earns 9% per year compounded monthly, what lump sum is equivalent?

Here \( r = 0.09/12 = 0.0075 \) and \( n = 240 \). First, \( (1.0075)^{240} = 6.009152 \), so \( (1.0075)^{-240} = 0.166413 \).

$$ PV = 50{,}000 \times \frac{1 - 0.166413}{0.0075} = 50{,}000 \times 111.144954 = \text{PKR } 5{,}557{,}247.70 $$

Any lump sum above about PKR 5.56 million would be better than the pension at this rate, assuming the officer can actually invest at 9%.

Deferred Annuities

A deferred annuity starts after a waiting period. Value it in two steps: find its value one period before the first payment using the ordinary formula, then discount that back to today.

Suppose PKR 100,000 is paid each year for 5 years, with the first payment at the end of year 4, at 10%. The annuity formula gives its value at the end of year 3:

$$ 100{,}000 \times 3.790787 = 379{,}078.68 $$

Discount three years to today:

$$ PV = \frac{379{,}078.68}{(1.1)^3} = \frac{379{,}078.68}{1.331} = \text{PKR } 284{,}807.42 $$

Perpetuities: Payments Forever

What if the payments never stop? Let \( n \to \infty \) in the annuity formula. Since \( (1+r)^{-n} \to 0 \),

$$ PV_{\text{perpetuity}} = \frac{PMT}{r} $$

It may seem strange that infinitely many payments have a finite value, but distant payments are discounted so heavily that they add almost nothing.

Example. A trust promises PKR 120,000 per year forever for student scholarships. At 8%, its value is

$$ PV = \frac{120{,}000}{0.08} = \text{PKR } 1{,}500{,}000 $$

Check the logic: PKR 1.5 million invested at 8% earns exactly PKR 120,000 per year, which can be paid out forever without touching the capital.

Growing Perpetuities

If the payment grows at a constant rate \( g \) each period (with \( g < r \)), and the first payment is \( PMT_1 \) one period from now:

$$ PV = \frac{PMT_1}{r - g} $$

If the scholarship grows by 5% a year to keep pace with fees, starting at PKR 120,000:

$$ PV = \frac{120{,}000}{0.08 - 0.05} = \text{PKR } 4{,}000{,}000 $$

This formula is the heart of the Gordon growth model used to value shares from their dividends.

Common Pitfalls

• Using an annual rate with monthly payments. The rate and payment frequency must match.
• Forgetting whether payments come at the start or end of the period.
• Applying the growing perpetuity formula when \( g \ge r \); the value is then infinite or meaningless.
• Using the current payment instead of next period's payment in \( PMT_1/(r-g) \).

Did You Know?

• Waqf endowments, common across the Muslim world for centuries, are a real-life perpetuity: the capital is preserved and only the income is spent, often supporting mosques, schools and hospitals.
• The British government issued perpetual bonds called consols; the last of them were redeemed in 2015, more than 250 years after the first were issued.
• At 10%, the first 30 years of a perpetuity account for about 94% of its total value.

Key Takeaways

• Ordinary annuity: \( PV = PMT\dfrac{1-(1+r)^{-n}}{r} \) and \( FV = PMT\dfrac{(1+r)^n-1}{r} \).
• For an annuity due, multiply the ordinary result by \( (1+r) \).
• Value deferred annuities in two steps: annuity value first, then discount.
• Perpetuity: \( PV = PMT/r \); growing perpetuity: \( PV = PMT_1/(r-g) \).
• Always match the rate to the payment period.

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