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.
A perpetuity is just an annuity that never ends. An annuity pays you for a fixed number of years; a perpetuity pays you forever.
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:
PV=rC
That's it. One division. No summation, no infinite series.
For our example: PV=0.10100=₹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:
PV=(1+r)C+(1+r)2C+(1+r)3C+⋯
This is an infinite geometric series. The first term is 1+rC, and the common ratio is 1+r1. For an infinite geometric series where ∣r∣<1, the sum is:
Sum=1−common ratiofirst term
So:
PV=1−1+r11+rC=1+rr1+rC=rC
›Proof
Derivation using algebra:
Let PV=1+rC+(1+r)2C+(1+r)3C+⋯
Multiply both sides by (1+r):
PV(1+r)=C+1+rC+(1+r)2C+⋯
Notice the right side is C+PV (since PV is the same infinite sum starting from 1+rC).
So: PV(1+r)=C+PV
PV+PV⋅r=C+PV
PV⋅r=C
PV=rC
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:
PV=C+rC
Because you get the first payment immediately, and then the standard perpetuity starts from next period.
A common mistake: students forget that the formula PV=C/r assumes payments start at the end of the first period, not today. If a problem says "payments start now," you must add the immediate payment.
Real-World Examples …