Finance-Theory · Unit 3 · Video 5 · Interactive Practice

What a Perpetuity's Price Tells You: Consols, Disney's 100-Year Bond, and Total Return

IKey Formulas

Formula	Name	What goes in, what comes out
$P = \dfrac{C}{r}$	Perpetuity price	A rate in, a price out: the level payment $C$ , first paid at date 1, discounted forever
$r = \dfrac{C}{P}$	Implied rate	The same equation inverted: a traded price in, the rate the market is using out
$R = \dfrac{(P_1 - P_0) + C}{P_0}$	Total return over a year	Price change plus coupon, over the price paid
$P_0 = P_1 = \dfrac{C}{r} \;\Rightarrow\; R = \dfrac{0 + C}{C/r} = r$	Flat price, full return	With $C$ and $r$ both fixed the price never moves and the coupon alone delivers $r$

Key Insight: One equation, two readings. Given the rate it prices the paper; given a traded price it reports the rate the market is using — which is how a rate was read off a consol for as long as consols traded. And because the price is the coupons discounted at $r$ , the coupon is exactly $r$ of the price: a perpetuity whose price never moves has still earned $r$ , all of it through the coupon.

IIPrice and Rate, Each Read Off the Other

A rate fixes the price; a traded price fixes the rate. One curve, entered from either axis.

Annual payment Price P $1,000 Rate r 10.0%

💡 Britain's consols were exactly this instrument — an undated bond, its price observable every day it traded — until the government redeemed its last four undated issues in full at par on 5 July 2015.

IIIA Hundred Payments Against Forever

Disney's bonds run to 2093, not forever. What share of a perpetuity do a hundred payments carry?

Payments counted 100 Rate r 5.0%

💡 A dated bond also repays its face value at maturity, and discounted at 5% over a hundred years that repayment is worth under a cent per dollar of face — it moves the comparison as little as the missing tail does.

IVA Flat Price That Still Earns r

The price never moves, and yet the holder earns $r$ every year. Where does it come from?

Rate (constant) Years held 0.0

💡 The zero price return depends on the rate staying put: if $r$ fell from 10% to 5%, the same $100 a year would be worth $2,000, and the holder of a $1,000 perpetuity would book a $1,000 capital gain on top of that year's coupon.

VQuiz Questions

Problem 1 · Read the Rate Off the Price

Given: a perpetuity pays $40 a year forever, the first payment a year from now, and it trades today at $500 — find the rate the market is using.

✅ Correct! Inverting

P = C/r

gives

r = C/P = 40/500 = 0.08

, so the buyer at $500 is accepting 8% a year.

❌ That ratio is upside down.

P/C = 500/40 = 12.5

says the price is worth twelve and a half years of payments; the rate is its reciprocal,

1/12.5 = 0.08

❌ Decimal slip.

C/P = 0.08

is already a fraction of the price, and

0.08 = 8\%

. Writing it as

0.08\%

divides by a hundred a second time.

❌ That is the video's payment, not this one.

100/500 = 20\%

prices a $100-a-year perpetuity; here

C = 40

, so

r = 40/500

Show solution

The perpetuity formula runs in either direction. Start from the price and solve for the rate:

P = \frac{C}{r} \quad\Longrightarrow\quad r = \frac{C}{P}

Substituting the payment and the traded price:

r = \frac{40}{500} = 0.08 = 8\%

Nothing was estimated here. The $500 is a price someone paid, and 8% is the rate that price implies — the market's rate, read off the tape.

Check it the other way: at $r = 0.08$ , $P = 40/0.08 = 500$ ✓

Problem 2 · The Winning Bid

Given: a certificate paying $1 a year forever is auctioned under a pay-your-bid rule (the highest bidder wins and pays their own bid). Sealed bids of $16, $24 and $32 arrive — find the rate the sale implies, and what the climbing bids do to the implied rate.

✅ Correct! The sale price is $32, so

r = C/P = 1/32 = 3.125\%

— and since

C

is identical in every bid, a higher price can only mean a smaller return.

❌ 6.25% belongs to the losing $16 bid.

1/16 = 6.25\%

is the rate that bidder wanted; the sale happened at $32, and $32 buys exactly the same $1 a year.

❌ The rate is right, the direction is backwards. In

r = C/P

the price sits in the denominator: $16 implies 6.25%, $24 implies 4.17%, $32 implies 3.125%. Paying more for the same payments means accepting less.

❌ That is the second-price rule. Under it the $32 bidder would win and pay $24, implying

1/24 \approx 4.17\%

— but this auction is pay-your-bid, so the price that clears is $32.

Show solution

Under a pay-your-bid rule the transaction price is the winning bid itself, $32 for $1 a year forever. Invert the perpetuity formula:

r = \frac{C}{P} = \frac{1}{32} = 0.03125 = 3.125\%

Run the same calculation on the other two bids to see the direction:

\frac{1}{16} = 6.25\%, \qquad \frac{1}{24} \approx 4.17\%, \qquad \frac{1}{32} = 3.125\%

The payment is identical in all three lines, so the price is the only thing moving — and it sits in the denominator. Every higher bid implies a lower rate: the winner pays more for the same stream and therefore accepts a smaller return. The video's auction runs the same way: bids of $5, $6 and $7 imply 20%, 16.7% and 14.3%.

The auction's design decides which bid becomes the price. A second-price rule would have settled here at $24, implying 4.17%; it is used because paying the runner-up's bid gives every bidder a reason to bid their true valuation.

Problem 3 · A Year in the Holder's Hands

Given: a perpetuity pays $60 a year, the first payment a year from now. The interest rate is 4% today and is still 4% a year from now.

The price today, and the price a year from now

The holder's total return over that year

✅ Correct! The price is

60/0.04 = 1500

at both dates, so the price change is zero and the whole return is the coupon:

60/1500 = 4\% = r

❌ That multiplied instead of dividing.

60 \times 0.04 = 2.40

is 4% of one payment; the price discounts the whole stream,

60/0.04

❌ Check the decimal. 4% is

0.04

, not

0.004

60/0.04 = 1500

, while

60/0.004

would be the price at a rate of 0.4%.

❌ The price does not grow with the coupon. A year on, the paper still promises $60 a year forever and the rate is still 4%, so

P_1 = 60/0.04 = 1500

again. The $60 is paid out to the holder, not added to the price.

❌ That is the price return only. The price really is flat, so

(P_1 - P_0)/P_0 = 0

— but the holder also banked a $60 coupon, and total return counts both.

❌ The coupon has been counted twice. The coupon yield

C/P_0

is the 4%; there is no separate 4% of price growth to add to it.

❌ Wrong base.

60/1560

divides by the year-end price plus the coupon. The return is measured on what the holder paid,

P_0 = 1500

Show solution

Step 1: price at each date. The payment is fixed by contract and the rate is unchanged, so the same formula gives the same number twice:

P_0 = \frac{60}{0.04} = 1500, \qquad P_1 = \frac{60}{0.04} = 1500

Step 2: the two pieces of the return.

R = \frac{(P_1 - P_0) + C}{P_0} = \frac{(1500 - 1500) + 60}{1500} = \frac{60}{1500} = 0.04 = 4\%

The price return is $0/1500 = 0$ : no capital gain at all. The coupon yield is $60/1500 = 4\%$ , and it is the entire return.

Why it must come out as $r$ . With $C$ and $r$ both fixed, $P_0 = P_1 = C/r$ , so the price change cancels and

R = \frac{0 + C}{C/r} = C \cdot \frac{r}{C} = r

The price is the coupons discounted at $r$ , so the coupon is exactly $r$ of the price. A flat price is not a failed investment — it is what earning $r$ entirely through the coupon looks like.

Problem 4 · A Century Instead of Forever

Given: a claim pays $50 a year for 100 years and nothing afterwards (no principal is repaid), at a rate of 8%. The matching perpetuity is worth $625, since $50/0.08 = 625$ . The video found that at 5% a hundred payments carry 99.2% of a perpetuity's value, and that a higher rate carries a larger share — find the value of the hundred-year claim.

✅ Correct! The missing tail is a perpetuity that starts in year 101, worth

625/(1.08)^{100} \approx 0.28

today — so the claim is worth about $624.72, or 99.95% of the perpetuity.

❌ Almost nothing is not nothing. Those payments are worth about 28 cents today — tiny, but the hundred-year claim is genuinely cheaper than the perpetuity.

❌ The share is not fixed. 99.2% is the figure at 5%. A higher rate discounts the distant payments harder, so at 8% the first hundred carry more than 99.2%, not exactly that.

❌ Value is not counted in years. The payments are weighted by their present values, and those collapse fast: at 8% the year-100 payment is worth about two cents, so the first hundred carry nearly the whole value.

Show solution

Price the missing part rather than the kept part. Everything from year 101 onward is a perpetuity of $50 a year that begins a hundred years from now, so it is worth $C/r$ as of year 100, discounted back:

\text{tail} = \frac{C/r}{(1+r)^{100}} = \frac{625}{(1.08)^{100}} = \frac{625}{2199.76} \approx 0.284

The hundred-year claim is the perpetuity minus that tail:

625 - 0.284 \approx 624.72 \quad\text{and}\quad \frac{624.72}{625} \approx 99.95\%

Compare the 5% case from the video, where the first hundred payments came to $19.848$ against a perpetuity value of $20$ , or 99.24%. The higher rate shrinks the far-off payments faster, so the truncated claim captures a larger share of the total.

The practical point: at ordinary rates, stopping the sum at a hundred terms costs a fraction of a percent. A century bond can be valued with the perpetuity formula, and the rate can be read back off its price with $r = C/P$ the same way.

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