Deriving the Price of a Perpetual Coupon Bond

Summary

The document derives the value of a perpetual bond, or consol, that pays a fixed annual coupon. Under discrete annual compounding, the bond’s price is the sum of each future coupon discounted by the corresponding power of one plus the required rate. Treating those discounted payments as a geometric series gives the familiar coupon-to-rate relationship.

A second explanation gives the same result through a replication-style intuition: an investment of one unit earning the required annual return produces that amount of income each year, so the number of coupon-paying bonds needed to generate it determines the price of each bond. The exchange clarifies that the stated formula uses discrete compounding. It assumes a constant coupon and rate, annual payments, and perpetual payment with no maturity or principal repayment; it does not discuss changing rates, credit risk, or alternative payment frequencies.

Key ideas

• A consol’s price can be derived by summing its indefinitely recurring discounted coupon payments.
• The discounted payments form a geometric series under a constant rate and annual payment schedule.
• The coupon-to-rate price formula corresponds to discrete annual compounding in this setup.
• The derivation assumes a constant coupon and rate and does not include principal repayment at maturity.

Tags

# Derive Perpetual Bond Price


# Derive Perpetual Bond Price












It is known that a perpetual bond with coupon $c$ has price $$P=\frac{c}{r}$$ How do you get to this price? Is $r$ stated in discrete or continuous compounding?

## Answer by Alex C (score 5, accepted)

https://quant.stackexchange.com/a/22297

A Consol Bond is a bond that pays an annual coupon of c every year. Therefore its price is $P=\frac{c}{1+r}+\frac{c}{(1+r)^2}+\cdots$. Factoring out the c and using the known formula for a geometric series, namely $u+u^2+u^3+\cdots = \frac{u}{1-u}$ we get $P=c[\frac{1}{1+r}/(1-\frac{1}{1+r})]=\frac{c}{r}$

Clearly this is a discrete compounding, not continous compounding formula.

## Answer by Mats Lind (score 2)

https://quant.stackexchange.com/a/45623

By definition; to get your required annual perpetual return of r, you trivially pay 1 USD up-front to get r USD annually. To get those annual payments from the consol bond in question you need to have r/c bonds (each paying c USD annually). To get those bonds for your 1 USD up-front payment, they have to sell at the price of c/r USD which is hereby demonstrated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.