Finance-Theory · Unit 3 · Video 7 · Interactive Practice

The Annuity: A Perpetuity on Borrowed Time

IKey Formulas

Formula	Name	What you need
$\text{PV} = \dfrac{C}{r} - \dfrac{C}{r}\dfrac{1}{(1+r)^T}$	$T$ -period annuity	$C$ , $r$ , $T$
$\text{PV} = \dfrac{C}{r}$	Perpetuity	First payment at date 1
$\dfrac{C}{r}\dfrac{1}{(1+r)^T}$	Date- $T$ perpetuity, valued at date 0	$T$ periods of discounting, not $T+1$
$r \times \text{PV} = C - \dfrac{C}{(1+r)^T}$	Multiply-and-subtract identity	The finite sum

Key Insight: An annuity is a perpetuity you buy at date 0 and sell at date $T$ . You pay $C/r$ and you are paid $C/r$ back — the only difference between the two terms is $T$ periods of discounting.

IIVisualization 1 — Building the Annuity by Subtraction

Two perpetuities, one of them deferred: everything after date $T$ cancels, and an annuity is what survives.

Build 1 Last payment date T 5

IIIVisualization 2 — Buy at Date 0, Sell at Date T

You sell the perpetuity back for the price you paid — how far does that price have to travel?

Discount the resale over

Date-0 ledger, in dollars	Amount
Buy the perpetuity at date 0	$-1{,}000.00$
Sell it at date 5, discounted five periods	$+620.92$
Net cost — what the five payments are worth	$379.08$

💡 Both routes agree: the algebra never mentions a buyer — dividing the multiply-and-subtract result by $r$ turns its two terms into the purchase price and the resale.

IVVisualization 3 — How Much of the Perpetuity Do T Payments Buy?

A perpetuity's value splits in two at date $T$ : the payments you keep and the ones you sell.

Last payment date T 5

VQuiz Questions

Problem 1 · Price a Four-Year Annuity

Given: $500 at the end of each of the next four years, with $r = 8\%$ — find the present value.

✅ Correct! The perpetuity costs $6,250 and the resale four years later returns $6,250, worth $4,593.94 today. The annuity is the gap between them.

❌ Close, but check the exponent. The resale happens at date 4, so it is discounted four periods; you used five.

❌ That is the perpetuity. $500 / 0.08 = $6,250 pays forever. This stream stops at date 4, so the value of everything from date 5 on must be subtracted.

❌ Not quite. The term you subtract is the whole resale price

C/r

discounted back, not a single payment of $500.

❌ Not quite. Value the perpetuity first, then subtract what it is worth from date 5 onward.

Show solution

Buy the perpetuity at date 0 and sell it the moment the date-4 payment arrives. The purchase price is

\frac{C}{r} = \frac{500}{0.08} = 6{,}250

Interest is constant, so the sale fetches the same price — but that is a date-4 value, brought home over four periods:

\frac{C}{r}\cdot\frac{1}{(1+r)^T} = \frac{6{,}250}{1.08^4} = \frac{6{,}250}{1.360489} = 4{,}593.94

The annuity is the net cost of the trade:

\text{PV} = 6{,}250 - 4{,}593.94 = 1{,}656.06

Check payment by payment: $462.96 + 428.67 + 396.92 + 367.51 = 1{,}656.06$ . The present value is $1,656.06.

Problem 2 · Date the Resale

Given: you buy a perpetuity paying $40 a year at $r = 5\%$ and sell it the moment the date-10 payment arrives — find the date-0 value of that sale.

✅ Correct! The price $800 is already dated at date 10, so ten periods of discounting bring it to date 0.

❌ Close — that is eleven periods. The buyer's first payment does arrive at date 11, but

C/r

is always the value one period before the first payment. Discounting eleven periods counts that year of waiting twice.

❌ That is nine periods. You hold the paper through the date-10 payment and hand it over then, so the sale is a date-10 value.

❌ $800 is the price at date 10, not at date 0. Ten periods of discounting are still to come.

❌ Not quite. Price the perpetuity at the sale date first, then discount that price back.

Show solution

At a constant interest rate a perpetuity's price never changes, so the buyer pays

\frac{C}{r} = \frac{40}{0.05} = 800

That is a date-10 value — the buyer's first payment comes at date 11, and $C/r$ sits one period before the first payment. So it travels back ten periods:

\frac{800}{1.05^{10}} = \frac{800}{1.628895} = 491.13

Check by discounting the buyer's payments directly and factoring out $1/1.05^{10}$ :

\frac{40}{1.05^{11}} + \frac{40}{1.05^{12}} + \cdots = \frac{1}{1.05^{10}}\left[\frac{40}{1.05} + \frac{40}{1.05^{2}} + \cdots\right] = \frac{1}{1.05^{10}}\cdot 800 = 491.13

The sale is worth $491.13 in date-0 dollars.

Problem 3 · Multiply and Subtract

Given: the finite sum below — multiply both sides by $1+r$ and subtract the original sum from the result.

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

What does the subtraction leave?

Now solve for PV.

✅ Correct! Two terms have no partner: the leading

C

of the multiplied sum and the original's last term

\frac{C}{(1+r)^T}

, which enters with a minus sign.

❌ Check the cancellation. Multiplying by

1+r

drops every exponent by one, so the new sum runs

C, \frac{C}{1+r}, \ldots, \frac{C}{(1+r)^{T-1}}

. Everything from

\frac{C}{1+r}

\frac{C}{(1+r)^{T-1}}

appears in both sums and cancels in pairs.

❌ Check the division. Dividing by

r

hits both terms:

C

becomes

\frac{C}{r}

and

\frac{C}{(1+r)^T}

becomes

\frac{C}{r}\frac{1}{(1+r)^T}

Show solution

Step 1 — Multiply by $1+r$ . Each denominator loses one power:

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

Step 2 — Subtract the original. The two sums share every term from $\frac{C}{1+r}$ through $\frac{C}{(1+r)^{T-1}}$ :

(1+r)\,\text{PV} - \text{PV} = C - \frac{C}{(1+r)^T} \qquad\Longrightarrow\qquad r \times \text{PV} = C - \frac{C}{(1+r)^T}

Step 3 — Divide by $r$ :

\text{PV} = \frac{C}{r} - \frac{C}{r}\frac{1}{(1+r)^T} = \frac{C}{r}\left(1 - \frac{1}{(1+r)^T}\right)

The perpetuity is the same argument with no last term: the series never ends, so only $C$ survives and $\text{PV} = \frac{C}{r}$ .

Problem 4 · Size a Loan Payment

Given: you borrow $20,000 at $r = 6\%$ and repay it with five equal annual payments, the first one year from now — find the payment $C$ .

✅ Correct! Five payments of $4,747.93 have a present value of exactly $20,000 at 6%.

❌ That compounds the whole loan forward. $20,000 × 1.06⁵ = $26,764.51 split five ways charges interest on principal you have already repaid.

❌ That repays the principal and nothing else. $20,000 / 5 ignores the interest owed on the balance outstanding each year.

❌ That is the perpetuity payment. $1,200 = 0.06 × $20,000 covers the interest forever and never repays the $20,000 itself.

❌ Not quite. Set the present value of five payments of

C

equal to the amount borrowed and solve for

C

Show solution

The repayments are a five-period annuity whose present value must equal the amount borrowed:

\frac{C}{r}\left(1 - \frac{1}{(1+r)^T}\right) = C \cdot \frac{1 - 1.06^{-5}}{0.06} = C \times 4.212364 = 20{,}000

C = \frac{20{,}000}{4.212364} = 4{,}747.93

Check it as a trade. A perpetuity paying $C = 4{,}747.93$ costs $C/r = 79{,}132$ , and reselling it at date 5 is worth $79{,}132 / 1.06^5 = 59{,}132$ today. The difference is the $20,000 you walked away with.

The payment is $4,747.93 a year.

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