Finance-Theory · Unit 3 · Video 4 · Interactive Practice

The Perpetuity: Why Cash Forever Is Worth Only C/r

IKey Formulas

Formula	Name	What it says
$\text{PV} = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots$	Perpetuity as a sequence	The general value formula with every cash flow set to the same $C$ and no last payment
$(1+r)\,\text{PV} = C + \text{PV}$	The multiply-by- $(1+r)$ step	Each term loses one factor of $(1+r)$ , so the series comes back with an extra $C$ in front
$r\,\text{PV} = C$	After the subtraction	Every term of the original series cancels against its twin; only the extra $C$ survives
$\text{PV} = \dfrac{C}{r} = \dfrac{1}{r} \times C$	Perpetuity formula	The value at date 0, one period before the first payment: $1/r$ payments' worth of cash

Key Insight: Each payment is worth $1/(1+r)$ of the one before it, so the terms shrink geometrically and infinitely many of them still add to the finite number $C/r$ — as long as the same $r$ sits in every denominator.

IIInfinitely Many Payments, a Finite Total

The payments never stop, yet the running total flattens against a ceiling — and that ceiling is $C/r$ .

Interest rate r 10% Payments counted n 12

IIIThe Trick: Multiply by $(1+r)$ , Then Subtract

Multiplying the series by $(1+r)$ hands back the same series with one extra $C$ in front.

Step 1

Step 1 — The series. Every cash flow equals $C$ , and there is no last one:

\text{PV} = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots \qquad (1)

Step 2 — Multiply both sides by $(1+r)$ . Every term loses one factor of $(1+r)$ , so its exponent drops by one:

(1+r) \times \frac{C}{(1+r)^t} = \frac{C}{(1+r)^{t-1}}

Step 3 — The new equation. The first term becomes the undiscounted $C$ , and everything after it is the original series again:

(1+r)\,\text{PV} = C + \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \cdots = C + \text{PV} \qquad (2)

Step 4 — Subtract (1) from (2). On the left, $(1+r)\,\text{PV} - \text{PV} = r\,\text{PV}$ . On the right, every term of (1) has an identical twin in (2), so the pairs cancel and only the extra $C$ is left.

Step 5 — What survives. The infinite series is gone from both sides:

r\,\text{PV} = C

Step 6 — Divide by $r$ .

\text{PV} = \frac{C}{r}

With $C = 100$ and $r = 0.10$ the paper is worth $1,000, and the check runs backwards: $r \times \text{PV} = 0.10 \times 1000 = 100 = C$ .

IVWhat Forever Is Worth: $1/r$ Payments

How much cash today does a claim to $100 a year forever trade for at the rate $r$ ?

Interest rate r 10% Your value 1500

💡 Two conditions hide inside that single number: the same $r$ must sit in every year's discount factor, and $C/r$ is the value at date 0 — one period before the first payment arrives.

VQuiz Questions

Problem 1 · Value a Perpetuity

Given: a piece of paper pays $450 a year forever, the first payment one year from today, and the interest rate is $r = 6\%$ . Find what the paper is worth today.

✅ Correct!

\text{PV} = C/r = 450/0.06 = 7500

, so the paper is worth $7,500 — about

1/0.06 = 16.7

payments of $450.

❌ That is only the first payment. $424.53 is

450/1.06

, the value of the cash arriving at date 1. Every later payment still has to be counted, and all of them together come to

C/r

❌ One discount too many. $7,075.47 is $7,500 divided again by

1.06

. The formula

C/r

already stands at date 0, one period before the first payment, so nothing further is discounted.

❌ That adds a payment today. $7,950 is

7500 + 450

, as if cash also arrived at date 0. The first payment is a full year away, so the series starts at

C/(1+r)

and sums to

C/r

exactly.

❌ Not quite. Divide the annual payment by the interest rate written as a decimal, not by

1+r

Show solution

The payments are equal, they never stop, and the first one arrives a full period from now — a perpetuity valued at date 0:

\text{PV} = \frac{C}{r} = \frac{450}{0.06} = 7500

The paper is worth $7,500.00 today.

Against the definition: the first payment is worth $450/1.06 = 424.53$ , the second $450/1.06^2 = 400.50$ , the third $450/1.06^3 = 377.83$ , and the shrinking terms add to exactly 7500.

Problem 2 · The Step That Sums the Series

Given: the equation $\text{PV} = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots$ is multiplied through by $(1+r)$ , and then the original equation is subtracted from the result. Find the equation that is left.

✅ Correct! The left side gives

(1+r)\,\text{PV} - \text{PV} = r\,\text{PV}

; on the right every term of the original series cancels against its twin, leaving the extra

C

alone.

❌ The subtraction has been skipped. Multiplying alone gives

(1+r)\,\text{PV} = C + \text{PV}

— the series is still there. Subtracting the original equation is what removes it.

❌ Check the first term. Multiplying

C/(1+r)

(1+r)

gives plain

C

: the exponent drops by one, so the term left over in front is the undiscounted payment.

❌ That keeps two terms of a series with no end. The whole point of subtracting is that every matched pair cancels at once, not that the series is cut off after a term or two.

❌ Not quite. Write out both equations term by term and line each term up under its twin before subtracting.

Show solution

Start from the series and multiply through, remembering that $(1+r) \times C/(1+r)^t = C/(1+r)^{t-1}$ :

(1+r)\,\text{PV} = C + \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \cdots = C + \text{PV}

Now subtract $\text{PV}$ from both sides:

(1+r)\,\text{PV} - \text{PV} = C \quad \Longrightarrow \quad r\,\text{PV} = C

The infinite series has disappeared from both sides, and dividing by $r$ gives $\text{PV} = C/r$ . The trick works only because the total is finite in the first place — the terms shrink by the factor $1/(1+r)$ every year.

Problem 3 · What a Change in the Rate Does

Given: an endowment pays $60 a year forever, the first payment one year from today. Find its value at $r = 8\%$ , and then its value at $r = 12\%$ .

Value at $r = 8\%$

Value at $r = 12\%$

✅ Correct! $750 at 8% and $500 at 12%: the multiple

1/r

fell from 12.5 payments to about 8.3, so the value fell by a third while the $60 payment never changed.

❌ Check the first value. The payment is divided by

r

itself — not multiplied by it, and not divided again by

1+r

, since

C/r

already stands at date 0:

60/0.08

❌ Check the second value. The same formula takes the new rate:

60/0.12

. The value does not fall by the 4 percentage points the rate rose — it is the reciprocal

1/r

that moves.

Show solution

Both values come from the same formula with the same $C = 60$ :

\text{PV}_{8\%} = \frac{60}{0.08} = 750 \qquad \text{PV}_{12\%} = \frac{60}{0.12} = 500

The endowment is worth $750.00 at 8% and $500.00 at 12%.

The value moves with $1/r$ , not with $r$ : multiplying the rate by $1.5$ divides the value by $1.5$ . Halving the rate would double the value instead — $60 at 4% is worth $1,500.

Problem 4 · Forever Against a Single Payment

Given: $r = 5\%$ . Paper A pays $40 a year forever, the first payment one year from today. Paper B pays $900 once, one year from today, and nothing afterwards. Find which is worth more today, and by how much.

✅ Correct!

40/0.05 = 800

against

900/1.05 = 857.14

: the single payment beats the endless one by $57.14, because at 5% forever is worth only 20 payments.

❌ Right gap, wrong direction. The two values are $800 and $857.14, so it is B that is worth more. An infinite stream is not automatically the larger number.

❌ B has not been discounted. $900 − $800 compares cash at date 1 with a value at date 0. B's payment is a year away, so it is worth

900/1.05 = 857.14

today.

❌ A has been discounted twice. $761.90 is

800/1.05

; but

C/r

is already the value at date 0, one period before A's first payment, so $800 is the number to compare.

❌ Not quite. Bring both papers to date 0 first: the perpetuity with

C/r

, the single payment by dividing once by

1+r

Show solution

Paper A is a perpetuity whose first payment is one period away, so $C/r$ is already its date-0 value:

\text{PV}_A = \frac{40}{0.05} = 800

Paper B is a single cash flow at date 1, divided by $1+r$ once:

\text{PV}_B = \frac{900}{1.05} = 857.142\ldots \approx 857.14

So B is worth more, by $857.14 - 800 = 57.14$ — $57.14.

At $r = 5\%$ the perpetuity is worth $1/0.05 = 20$ payments of $40, which is $800; one payment of $900 next year beats it. Forever buys a multiple of the payment, and when $r$ is high that multiple is small.

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The Perpetuity: Why Cash Forever Is Worth Only C/r

IKey Formulas

IIInfinitely Many Payments, a Finite Total

IIIThe Trick: Multiply by (1+r)(1+r)(1+r), Then Subtract

IVWhat Forever Is Worth: 1/r1/r1/r Payments

VQuiz Questions

IIIThe Trick: Multiply by $(1+r)$ , Then Subtract

IVWhat Forever Is Worth: $1/r$ Payments