Q.Find the present value of a sequence of payments of ₹80 made at the end of each 6 months and continuing forever, if money is worth 4% compounded semi-annually.

Present Value of a Perpetuity: The Intuition

Imagine you own a magic tree that gives you exactly ₹100 every year, forever. No matter what happens, that ₹100 arrives each year, year after year, for eternity. What would you sell that tree for today? That's the question the present value of a perpetuity answers.

The key insight is simple: a rupee today is worth more than a rupee tomorrow. So those future ₹100 payments, when brought back to today, are worth less than ₹100 each. The further away the payment, the less it's worth today.

If you think about it, the first ₹100 (one year from now) is worth about ₹91 today (at 10% interest). The second ₹100 (two years from now) is worth about ₹83 today. The third is worth about ₹75. And so on, getting smaller and smaller.

Now here's the beautiful part: if you add up all these shrinking values — the ₹91, the ₹83, the ₹75, and so on forever — they don't add up to infinity. They converge to a finite number. That finite number is the present value of the perpetuity.

The Precise Statement

Let:

• $C$ = the constant cash flow received each period (e.g., ₹100 per year)
• $r$ = the discount rate per period (e.g., 10% = 0.10)

The present value $PV$ of a perpetuity is:

That's it. One division. No summation, no infinite series.

For our example: $PV = \frac{100}{0.10} = ₹1000$.

So the magic tree is worth ₹1000 today. Why? Because if you put ₹1000 in a bank account earning 10% per year, you'd get ₹100 in interest each year — forever — without ever touching the principal. The tree and the bank account produce the same cash flow, so they have the same value.

Where Does This Formula Come From?

The present value is the sum of all future cash flows discounted back to today:

This is an infinite geometric series. The first term is $\frac{C}{1+r}$, and the common ratio is $\frac{1}{1+r}$. For an infinite geometric series where $|r| < 1$, the sum is:

So:

The Critical Assumption

The formula $PV = C/r$ assumes the first payment occurs one period from today (an "ordinary perpetuity"). If the first payment occurs today (a "perpetuity due"), the formula becomes:

Because you get the first payment immediately, and then the standard perpetuity starts from next period.

Real-World Examples

1. Preferred stock: Some companies issue preferred shares that pay a fixed dividend forever. If a share pays ₹5 per year and the required return is 8%, its value is $\frac{5}{0.08} = ₹62.50$.

2. Consols: The British government issued bonds called "consols" that paid interest forever with no maturity date. Their price was simply the annual coupon divided by the yield.

3. Perpetual scholarships: A university wants to fund a ₹1,00,000 annual scholarship forever. If the endowment earns 5%, they need $\frac{1,00,000}{0.05} = ₹20,00,000$ today.

Growing Perpetuities

Sometimes the cash flow grows at a constant rate $g$ each period (e.g., a dividend that grows 3% per year). The formula becomes:

This is the Gordon Growth Model, widely used in stock valuation. It requires $r > g$; otherwise, the value is infinite (the growth outpaces the discounting).

Key Takeaways for Exams

• Memorize: $PV_{\text{perpetuity}} = \frac{C}{r}$
• Check timing: Is the first payment at the end of period 1 or at time 0?
• Growing version: $PV = \frac{C}{r - g}$ (only valid when $r > g$)
• Intuition check: The perpetuity value equals the amount of money you'd need to invest today at rate $r$ to generate $C$ each period forever.

The beauty of the perpetuity formula is its simplicity. One division captures the value of an infinite stream of payments — a powerful idea that appears in everything from bond pricing to company valuation.